[Grundlehren der mathematischen Wissenschaften] Proof Theory Volume 225 ||

Grundlehren der mathematischen Wissenschaften 225

A Series of Comprehensive Studies in Mathematics

Editors

S. S. Chern J. L. Doob J. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov W. Schmidt D. S. Scott K. Stein J. Tits B. L. van der Waerden

Managing Editors

B. Eckmann J. K. Moser

Kurt Schutte

Proof Theory

Translation from the German by J. N. Crossley

Springer-Verlag Berlin Heidelberg New York 1977

Kurt Schutte

Mathematisches Institut, Ludwig-Maximilians-Universitat, 8000 Munchen 2/Germany

Translator:

J. N. Crossley

Department of Mathematics, Monash University, Clayton, Victoria 3168/ Australia

Translation of the revised version of "Beweistheorie", 1 st edition, 1960; Grundlehren der mathematischen Wissenschaften, Band 103

AMS Subject Classification (1970): 02B 10, 02B99, 02C 15, 02Dxx, 02E99, ION99

ISBN-13: 978-3-642-66475-5 DOl: 10.1007/978-3-642-66473-1

e-ISBN-13: 978-3-642-66473-1

Library of Congress Cataloging in Publication Data. Schutte, KUTt. Proof theory. (Grundlehren def

mathematischen Wissenschaften; 225). Translation of Beweistheorie. Bibliography: p. Includes index. I. Proof theory. I. Title. II. Series: Die Grundlehren dec mathematischen Wissenschaften in Einzeldarstellungen; 225. QA9.54.S3813. 511'.3. 76-45768. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation. reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under §54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1977. Softcover reprint of the hardcover 1st edition 1977

2141/314()-543210

Preface

This book was originally intended to be the second edition of the book "Beweistheorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic.

We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie".

My hearty thanks are due to Professor J. N. Crossley for his translation of my German manuscript and for his careful checking of the proof sheets. I also wish to thank Dr. W. Pohlers and Dr. W. Buchholz for the help which they have given me technically and also in reading the proofs. I would especially like to thank Frau Ernst (Munich) and Ms. Vandenberg (Monash University) for their very careful typing of the text. I wish to thank Springer-Verlag for their kindness and help in the production of the book.

Munich, March 1977 Kurt Schutte

Table of Contents

Introduction . 1

Part A. Pure Logic 5

Chapter 1. Fundamentals 7

§ 1. Classical Sentential Calculus . 7 1. Truth Functions 7 2. Sentential Forms . 8 3. Complete Systems of Connectives 8 4. A Formal Language for the Sentential Calculus 9 5. Positive and Negative Parts of Formulas 10 6. Syntactic Characterization of Valid Formulas 12

§ 2. Formal Systems 13 1. Fundamentals . 13 2. Deducible Formulas 14 3. Permissible Inferences . 15 4. Sentential Properties of Formal Systems. 16 5. The Formal System CS of the Classical Sentential Calculus 17

Chapter II. Classical Predicate Calculus 19

§ 3. The Formal System CP . 19 1. Primitive Symbols . 19 2. Inductive Definition of the Formulas 20 . 3. P-Forms and N-Forms 20 4. Positive and Negative Parts of a Formula 20 5. Axioms. 20 6. Basic Inferences 20

§ 4. Deducible Formulas and Permissible Inferences 21 1. Generalizations of the Axioms 21 2. Weak Inferences 22 3. Further Permissible Inferences 24 4. Defined Logical Connectives. 25

VIn

§ 5. Semantics of Classical Predicate Calculus 1. Classical Models 2. The Consistency Theorem. 3. The Completeness Theorem 4. The Satisfiability Theorem 5. Syntactic and Semantic Consequences

Chapter III. Intuitionistic Predicate Calculus .

§ 6. Formalization of Intuitionistic Predicate Calculus. 1. The Formal System IPI 2. The Formal System IP2 3. Left and Right Parts of Formulas 4. The Formal System IP3

Table of Contents

26 27 28 28 32 35

36

36 36 38 39 40

§ 7. Deducible Formulas and Permissible Inferences in the System IP3 42 1. Generalizations of the Axioms 42 2. Weak Inferences 42 3. More Permissible Inferences . 44 4. Special Features of Intuitionistic Logic 46 5. Properties of Negation. '. 47 6. Syntactic Equivalence . 47

§ 8. Relations between Classical and Intuitionistic Predicate Calculus 48 1. Embedding IP3 in CP . 48 2. Interpretation of CP in IP3 49

§ 9. The Interpolation Theorem . 1. Interpolation Theorem for the System IP3 2. Interpolation Theorem for the System CP 3. Finitely Axiomatisable Theories 4. Beth's Definability Theorem .

Chapter IV. Classical Simple Type Theory.

§ 10. The Formal System CT 1. The Formal Language. 2. Chains of Subterms 3. Axioms and Basic Inferences 4. Deducible Formulas and Permissible Inferences 5. The Cut Rule

§ 11. Deduction Chains and Partial Valuations 1. Definition of Deduction Chains 2. Partial Valuations . 3. Principal Lemmata.

§ 12. Semantics . 1. Total Valuations over a System of Sets 2. Soundness Theorem

51 51 54 54 55

56

56 56 57 60 60 62

62 62 63 64

65 65 66

Table of Contents

3. Extending a Partial Valuation 4. Completeness Theorem and Cut Rule

Part B. Systems of Arithmetic

Chapter V. Ordinal Numbers and Ordinal Terms.

IX

67 69

71

73

§ 13. Theory of Ordinals of the 1st and 2nd Number Classes 73 1. Order Types of Well-Ordered Sets 73 2. Axiomatic Characterization of the 1st and 2nd Number Classes 74 3. Zero, Successor and Limit Numbers and Supremum 75 4. Ordering Functions 76 5. Addition of Ordinals . 79 6. a-Critical Ordinals . 81 7. Maximal a-Critical Ordinals 84

§ 14. A Notation System for the Ordinals <To 1. Definition of Ordinal Terms . 2. The Ordering of the Ordinal Terms 3. Addition of Ordinal Terms 4. Ordinal Terms cPaf3. 5. Ordinal Terms wa, wa·n and wa.f3 . 6. Ordinal Terms n·f3, 2a and 2a .f3 7. Ordinal Terms Ca' wn(a) and (n 8. A Mapping onto the Natural Numbers

Chapter VI. Functionals of Finite Type

§ 15. The System of Terms of Finite Type. 1. Types 2. Terms 3. Interpretation 4. Reduction Procedure for Terms 5. Characterization of Numerals 6. Substitution Properties 7. The Normal Form of a Term .

§ 16. Orders of Terms 1. Natural Sum and Natural Product of Ordinal Terms 2. Assigning Ordinal Terms to Terms 3. Estimates of the Order of Terms . 4. Equality of Terms .

§ 17. The Formal System FT of Functionals of Finite Type. 1. The Formal Language. 2. Deduction Procedures . 3. The Consistency of the System FT 4. Fundamental Deduction Rules

86 86 88 89 90 92 94 95 96

98

98 98 98 99

100 101 101 104

105 105 106 108 112

113 113 114 115 117

x Table of Contents

5. Addition and Multiplication . 120 6. The Indentity Functional I, and A-Abstraction . 121 7. The Predecessor Functional and the Arithmetic Difference 122 8. The Recursor . 124 9. Simultaneous Recursion . 126

10. The Characteristic Term of a Basic Formula 129

Chapter VII. Pure Number Theory. 134

§ 18. The Formal System PN for Pure Number Theory. 134 1. The Formal Language. 134 2. The Deduction Procedure. 135 3. Basic Properties of Deducibility 136 4. Properties of Negation. 138 5. Positive and Negative Parts of Formulas 142 6. The Consistency of the System PN . 147

§ 19. Interpretation of PN in FT . 148 1. Sequences of Terms of the System FT 148 2. The Formal System QFT . 149 3. Interpreting Formulas. 155 4. Interpretations of the Axioms of the System PN 157 5. Interpretations of the Basic Inferences in the System PN 158

Part C. Subsystems of Analysis . 165

Chapter VIII. Predicative Analysis. 167

§ 20. Systems of Ll~-Analysis 167 1. The Formal Language of Second Order Arithmetic 167 2. The Formal System DA . 170 3. Deducible Formulas and Permissible Inference of the System DA 171 4. The Semi-Formal System DA * 174 5. Embedding DA in DA * 175 6. General Properties of Deduction in the System DA * 176 7. Subsystems ofDA and DA* . 176

§ 21. Deductions of Transfinite Induction. 177 1. Formalisation of Transfinite Induction 177 2. Deductions in EN . 179 3. Deductions in EN*. 181 4. Deductions in EA and EA* 182 5. The Formula ~ [P, Q, t] 184 6. Deductions in DA . 188 7. Deductions in DA* 192

§ 22. The Semi-Formal System RA* for Ramified Analysis. 197 1. The Formal Language. 197

Table of Contents Xl

2. The Deduction Procedures 200 3. Weak Inferences 201 4. Elimination of Cuts 201 5. Further Properties of Deductions 204 6. Interpretations of EA * and DA * in RA * 207

§ 23. The Limits of the Deducibility of Transfinite Induction 209 1. Orders of Deductions of Induction in RA * . 209 2. The Limiting Numbers of the Systems EN, EA and DA 211 3. The Autonomous Ordinal Terms of the Systems EN*, EA* and

DA* . 213 4. The Autonomous Ordinal Terms of the System RA * 217 5. The Limits of Predicativity 220

Chapter IX. Higher Ordinals and Systems of 1I~-Analysis 221

§ 24. Normal Functions on a Segment 0* of the Ordinals 221 1. Axiomatic Characterization of the Segment 0* of the Ordinals 221 2. Basic Properties of 0* . 222 3. Definition of the Functions (Jrx 224 4. Properties of the (J Functions . 225 5. The Sets In(a) and Functions {}rx 230

§ 25. A Notation System for Ordinals Based on the (Jrx Functions 234 1. The Set {}(Q) of Ordinals . 234 2. Sets of Coefficients . 234 3. The Systems T* and OT* of Terms 236 4. Subsystems {}(r) of {}(Q) 238 5. The Ordinal Ao . 239 6. Relations between Cr (a) and In (a) 240

§ 26. Level-Lowering Functions of the Ordinals. 241 1. Basic Concepts . 242 2. Properties of the Sets of Coefficients . 242 3. The Ordinal Term dia 245 4. The Natural Sum . 248 5. Deduction Functions 248

§ 27. The Formal System GPA for a Generalized 1I~-Analysis . 252 1. The Formal Language. 252 2. Axioms, Basic Inference and Substitution Inferences 253 3. Deductions. 255 4. Orders of Normal Deductions 257 5. Transformations of Normal Deductions. 258 6. Reducible Normal Deductions 259 7. Singular Normal Deductions. 261 8. Reduction of a Suitable Cut . 262 9. The Consistency of the System GPA. 268

10. The Subsystem PA of GPA . 268

xu Table of Contents

§ 28. The Semi-Formal System PA* 269 269 270 271 272 273 275 277 278

I. Axioms and Basic Inferences of the System PA * 2. The Strength of a Formula 3. Basic Deductions in the System PA*. 4. Embedding ofPA in PA* . 5. Elimination of Strong Cuts in PA * . 6. Normal Deductions in the System PA* 7. Reducible Normal Deductions 8. Elimination of Cuts in PA*

§ 29. Proof of Well-Ordering . 281 I. A Constructive Proof of Well-Ordering for Subsystems of B(Q) 281 2. The Formal System IDn of n-Fold Iterated Inductive Definitions. 285 3. Formalization of the Proof of Well-Ordering of B(N) in IDN 287 4. Embedding IDn in a Subsystem of PA .. 289

Bibliography . 293

Subject Index . 297

Introduction

In mathematics, dealing with infinite number systems and other infinite sets brings a fundamental problem with it. For example consider the two statements:

I. Every even number greater than 2 can be represented as the sum of two primes.

II. There is an even number which is greater than 2 but cannot be represented as the sum of two primes.

According to classical logic the second statement is the contradictory of statement I. So far neither of these two statements can be proved. We cannot prove statement I by testing its validity for all even numbers since there are infinitely many such numbers. In order to prove statement I we must rather solve the following problem:

(I) To give a general procedure which assigns to each even number g > 2 two primes peg) and q(g) such that the equation

g = peg) + q(q)

is satisfied for all g > 2. If one does not succeed in solving this problem then one must attempt to prove

the opposite statement, II. Then one is dealing with an existential statement. If one takes the existential question seriously then in order to prove statement II one must solve the following problem:

(II) To give an even number go> 2 which one can recognize as not being representable as the sum of two prime numbers.

We now ask whether either of the two statements I and II is fundamentally correct. Now the correctness of a mathematical statement can only be determined by a proof. There is no sense in accepting a mathematical truth which is not possibly provable. So we can only ask whether one of statements I and II is provable in principle. If one says this question is solvable in the affirmative then one must be certain that one of problems (I) or (II) really is solvable. But such certainty is not attainable. In fact one cannot even deduce from the assumption that problem (I) is unsolvable that a number go, which satisfies problem II, can actually be shown to exist. Neither can one deduce, from the assumption that problem (II) is unsolvable, that there is a general procedure for solving problem (I).

2 Introduction

We therefore have no grounds for assuming that one of the two statements I and II is correct. The assumption would only be justified if a proof of I or II were given. But then the general situation regarding the provability of mathematical propositions would not be altered since there are always other cases where the question Of provability of a statement or its contradictory remains open. Therefore there is no immediate sense in saying that the principle of Tertium non datur, which says that either a proposition or its logical contradictory is correct, be accepted as satisfactory for mathematical propositions.

This brief critique of Tertium non datur leads us to the basic problem of proof theory. We start from the premise that in mathematics Tertium non datur is not meaningful as a general basic principle. But if one gives up this principle as a method of proof then one loses the right to indirect proof procedures such as one cannot do without in classical mathematics. In order to guarantee the correctness of classical mathematics in a logically unobjectionable way there only remains the possibility of basing mathematical statements not on provability as such hut on formal deducibility. This deducibility which may be regarded as a notional provability must be given axiomatically so that it is sufficient for the basic theorems of classical mathematics. In this way one does not need to interpret the concept of deducibility in a meaningful way, but only to establish its formal admissibility by means of a consistency proof. This is in fact Hilbert's programme: to establish the consistency of mathematics in its classical form.

Hilbert called the theory which solves this problem proof theory (Beweistheorie). The object of a proof-theoretic investigation is a mathematical theory which is precisely determined not only by its mathematical notions and axioms but also by the basic logical concepts and modes of inference which it allows. Now one must distinguish clearly between the laws of the theory which are to be investigated and those of the theory in which the investigation is to be carried out. The notions and theorems of the theory under investigation are represented in a formal system and treated purely formally without reference to their meaning, while the prooftheoretic investigation is concerned with the logical meaning of the notions and inference modes. Thus the formal theory is complemented by a meaningful metatheory (proof theory). This metatheory is also called "metalogic" or "metamathematicl'. This does not mean that one is extending the usual logic or mathematics, as/ts the case with metaphysics and physics. Here it only delineates the distinction between formalized mathematics and the logic in which the investigation is carried out.

If proof theory is to provide a foundation for a problematic area of mathematics then its own methods and notions and modes of inference must be restricted to those which lie outside this problematic area. The methods which Hilbert called "finitist" (fin it) do lie outside. These consist of just elementary logical and mathematical relations of a combinatorial nature which are used as bases for all theoretical investigations and are generally acknowledged as admissible. In a finitist proof Tertium non datur may not be used for propositions concerning infinite totalities and existential propositions are only to be treated as valid if the object whose existence is asserted can in fact be constructed.

But GOOel's investigations [1] have shown that the strictest finitist methods are

Introduction 3

basically inadequate for carrying out the consistency proof required by Hilbert's programme. So proof theory needs not only the very strict finitist methods of a combinatorial nature but also higher level proof procedures. Thus we arrive at methods, first used by Gentzen [2], using induction which in fact goes beyond the usual complete (mathematical) induction but still has a constructive character.

In this book we give a systematic presentation of the most important results which have so far been achieved in the pursuit of Hilbert's programme. The main emphasis is on consistency proofs but other investigations, which are closely related in terms of methodology, are also carried out. We use inductive methods for the consistency proofs but do not admit Tertium non datur as a proof procedure. We need to forgo the constructive standpoint and use classical mathematics as a foundation despite its problems only for the semantic treatment of classical predicate calculus (§5) and classical simple type theory (§12). Likewise we first consider the necessary systems of ordinals (in §13 and §24) from the standpoint of classical mathematics but then (in §§14, 25 and 26) we define these and treat them constructively.

As in the first edition of this book we use the notions of "positive and negative parts" as a basis for formalization in the realm of classical logic. These are a sort of generalization of the notions of "succedent and antecedent formulas" of Gentzen's sequence calculus (Sequenzenkalkul). In intuitionistic logic we use instead the weaker notions of "left and right parts of a formula".

In Chapter I we collect together the general basic syntactic notions which are n<teded in the book and used for the formalization of classical propositional calculus.

Chapter II treats the constructive syntax (§§3, 4) and the non-constructive semantics (§5) of classical predicate calculus. Intuitionistic predicate calculus is developed in Chapter III but only syntactically (constructively) and related to classical predicate calculus. The Interpolation Theorem proved here (§9) also yields the corresponding theorem for classical predicate calculus.

In Chapter IV we first assemble the basic syntactic properties for classical simple type theory in §IO and then introduce partial valuations in §11 which are used to carry out the non-constructive proof, following M. Takahashi [I] and D. Prawitz [1], of Takeuti's Fundamental Conjecture (Eliminability of Cuts).

In Chapter V we introduce a notation system for a segment of the ordinals which is used later for the treatment of pure number theory (Chapter VI) and predicative analysis (Chapter VIII). This system of notations is defined in §13 in terms of an ordinary non-constructive approach in the classical theory of ordinal numbers and then defined as a constructive system of ordinal terms in §14. In this section too we constructively prove the necessary properties of functions on ordinal terms of this system (which are needed later) in an elementary way.

In Chapters VI and VII we develop the Godel interpretation of pure number theory in a system offunctionals of finite type. Following W. Howard we prove (in §16) by transfinite induction up to 60 that the functionals introduced in §15 are calculable. In §17 we introduce a formal system FT for these functionals in the context of the positive implicational calculus. To this end Chapter VII treats, in

4 Introduction

§18, an obvious formal system of pure number theory, and in §19 we carry out the interpretation of this system in the .system FT.

In Chapter VIII we are concerned with the predicative limits of analysis and first we develop a formal and a semi-formal system of Ll~-analysis whose languages are that of classical second order arithmetic and which have restricted comprehension axioms as opposed to classical mathematics. In §21 we give deductions of formalized transfinite induction in the spirit of S. Feferman in these systems and in subsystems of Ll ~-analysis. In §22 we introduce a semi-formal system for ramified analysis in which the system of Ll~-analysis can be interpreted. This yields the limits for the deducibility of transfinite induction for the formal and semiformal systems considered and demonstrates the predicativity of these systems.

In Chapter IX we prove the consistency of a formal system of lIt-analysis following a proof ofG. Takeuti [4]. For this we need a transfinite induction which goes far beyond those previously introduced. Here we do not use the ordinal diagrams of Takeuti but a system of notations for ordinal terms based on normal functions ()~ defined and studied by S. Feferman, P. Aczel and J. Bridge and constructively characterized by W. Buchholz. Using these functions ()~ which are introduced in §24 in the context of the classical theory of ordinals, we build a constructive system of notations in §25. The necessary properties of this system of notations which are needed for the consistency proof are developed in §§25 and 26. §27 gives the consistency proof for the formal system of lIt-analysis and essentially this corresponds to the proof of Take uti but in a somewhat different way. Similarly we give a consistency prooffor a semi-formal system of lIt-analysis in §28, but this is done using a weaker transfinite induction than in §27. Finally we give constructive proofs of well-ordering for the systems of ordinals which we have used and their subsystems in §29.

Part A

Pure Logic

Chapter I

Fundamentals

In mathematical logic sentences are represented by formulas. Those investigations which only depend on the formal structure of logical formulas and fixed formal rules of deduction are said to be syntactic. On the other hand those which depend on model-theoretic interpretations are said to be semantic. Syntactic investigations, which feature predominantly in proof theory, are usually carried out constructively whilst the semantic ones can only be carried out with constructive concepts and proof procedures in the simplest part of logic, that is, in the sentential (or propositional) calculus.

In § I we start with a semantic characterization of classical sentential calculus and then develop an adequate syntax for it. In §2 we present the basic syntactic concepts which will be used in the subsequent chapters.

§ 1. CJassical Sentential Calculus

1. Truth Functions

Truth values are denoted by t (true) andf(false). An n-place truth function is an assignment of one truth value (t or f) to each n-tuple of truth values. Classical sentential calculus can be presented semantically as the theory of these truth functions.

As there are exactly 2" distinct n-tuples of truth values and each n-place truth function assigns just one of two possible values to each n-tuple there are exactly 2(2") distinct n-place truth functions.

We use the following connectives as symbols for truth functions. T (verum) and .l. (falsum) denote the O-place truth functions with values T = t and .l. = f

The connective. (negation) denotes the one-place truth function with values .(t)=fand '(f)=t.

The connectives v (disjunction), 1\ (conjunction) and - (implication) denote the 2-place truth functions with values as follows:

vet, t)= v (t,f)= v (I, t)=t, v(f,f)=f, I\(t, t)=t, I\(t,f)= 1\(1, t)= I\(f,f) =1, -(t, t) =-(1, t) =-(f,f) = t, -(t,f) = f

8 I. Fundamentals

In the same way we could define arbitrarily many other connectives. But they will not be used in the sequel.

2. Sentential Forms

Using connectives and sentential variables, which we denote by v, VI' V 2 , ••• , we construct sententialforms according to the following inductive definition:

SF!. Every sentential variable is a sentential form. SF2. Every O-place connective is a sentential form. SF3. If ¢ is an n-place connective (n ~ I) and A I, ... , An are sentential forms

then ¢(A 1> ••• , An) is also a sentential form. Such an inductive definition, like the ones we use constantly in the sequel, is to

be interpreted as follows: a string of symbols has the inductively defined property if, and only if, it can be obtained by a finite number of applications of the defining rules.

Example. -( /\ (VI' v2), v (1.., ,(VI») is a sentential form since the defining rules SFI-SF3 yield the following sentential forms:

I. VI and V 2 by SFI, 2. 1.. by SF2, 3. ,(vd by SF3 from I, 4. /\ (VI' v2 ) by SF3 from I, 5. v(1.., ,(VI» by SF3 from 2 and 3, 6. -( /\ (VI' v2 ), v (1.., ,(VI») by SF3 from 4 and 5.

The semantics of the sentential calculus are determined by sentential valuations. Such a valuation is an assignment V of a truth value Vv to each sentential variable v.

For a given valuation Vevery sentential form A gets a truth value V A according to the following inductive definition.

V I. A sentential form which is a sentential variable valone gets the truth value Vv given by the valuation V.

V2. VT=t, V1..=! V3. If ¢ is an n-place connective (n~ I) and AI' ... , An are sentential forms,

then V¢(A I, ... , An) is the truth value which the truth function represented by ¢ assigns to the n-tuple of truth values (V A I, ... , VAn).

Example. For a valuation V where VV I =f and VV2 =t we have V - (/\ (VI' v2), v (1.., ,(VI))) =-( /\ (f, t), v (f, ,(f)) =-(f, v (f, t» =-(f, t) = t.

:Two sentential forms A and B are said to be semantically equivalent if VA = VB for every sentential valuation V.

3. Complete Systems of Connectives

A set M of connectives is said to be a complete system of connectives if to every

1. Classical Sentential Calculus 9

sentential form A there is a semantically equivalent sentential form Ball of whose connectives are in M.

Theorem 1.1. The connectives --', /\ and v form a complete system of connectives.

Proof Let A be an arbitrary sentential form. We prove by induction on the number of sentential variables in A that there is a sentential form B, semantically equivalent to A, in which the only connectives are --', /\ and v .

I. A contains no sentential variables. Then A has the same truth value V A for every valuation V. If VA = t, then A is semantically equivalent to the sentential form v (v, --, (v». If VA =f, then A is semantically equivalent to the sentential form /\ (v, --, (v».

2. A contains the sentential variables VI' ••• , Vn (n~I). Let Al (respectively, . A 2) be the sentential form obtained from A by replacing Vn by T (respectively, 1-). By the induction hypothesis there are sentential forms B1 , B2 semanticallyequivalent to AI' A2 in which the only connectives are --', /\ and v. But then A is semantically equivalent to the sentential form v (/\ (BI' vn), /\ (B2' --, (vn ))).

Corollary 1.2. The connectives --, and v form a complete system of connectives.

Proof Immediate from Theorem 1.1 since every sentential form /\ (A, B) is semantically equivalent to the sentential form --, (v (--, (A), --, (B»).

Corollary 1.3. The connectives --, and -4 form a complete system of connectives.

Proof Immediate from corollary 1.2 since every sentential form v (A, B) is semantically equivalent to the sentential form -4(--' (A), B).

Corollary 1.4. The connectives 1- and -4 form a complete system of connectives.

Proof Immediate from Corollary 1.3 since every sentential form --, (A) is semantically equivalent to the sentential form -4(A, 1-).

There are many other complete systems of connectives but we shall not go into them as we shall not use them in the sequel.

4. A Formal Language for the Sentential Calculus

In formalizing classical sentential calculus we need only consider connectives from a complete system of connectives since semantically equivalent sentential forms have the same logical interpretation. We now define a formal language with connectives 1-, --', /\, v and -4. These connectives certainly form a complete system but of course they do not form a minimal complete system of connectives.

We assume we are given a countably infinite set of symbols which we call sentential variables. In fact we shall never need to consider these symbols themselves. We shall only use syntactic symbols v, VI' V2,'" for the given symbols for

10 I. Fundamentals

sentential variables. Theformulas of our formal language are given by the following inductive definition.

Fl. Every sentential variable is aformula. F2. 1.. is aformula. F3. If A is a formula then IA is aformula. F4. If A and B are formulas then (A 1\ B), (A v B) and (A - B) are also

formulas. Formulas may be identified with particular sentential forms where IA stands

for I (A) and the 2-place connectives are placed between rather than in front of their arguments following ordinary usage. The formulas 1.., lA, (A 1\ B), (A v B) and (A _ B) are to be read "falsum", "not A", "A and B", "A or B" and "A implies B".

For any valuation V every formula gets a truth value as defined for sentential forms. Accordingly we have here the following particular defining rules for V2 and V3 (from page 8):

V2. V 1..=f, V3.1. If VA = t (respectively, VA = f), then V,A = f(respectively, V,A = t). V3.2. If VA = t and VB= t, then V(A 1\ B) = t. Otherwise V(A 1\ B) = f V3.3. If VA = tor VB= t, then V(A v B) = t. Otherwise V(A v B) = f V3.4. If VA=for VB=t, then V(A- B)=t. Otherwise V(A-B)=f A formula A is said to be valid if VA = t for every sentential valuation V. Given a formula A we can decide, in an elementary way, whether it is valid or

whether there is a valuation under which it gets the value f Indeed there are only finitely many possible combinations of truth values for the sentential variables occurring in A and in order to decide whether A is valid we only have to compute V A for these.

Our aim now is to characterize the valid formulas just by their structure and without considering valuations and then to employ such a syntactic characterization again for stronger logical systems where the semantics cannot be expressed in as elementary and constructive a way as they can for classical sentential calculus.

5. Positive and Negative Parts of Formulas

A formula A which is part of a formula Fmay have the property that the truth value VFis already determined by the truth value VA. We shall call A apositive(negative) part of Fif VA =t (VA =f) implies VF=t.

Examples. A is a positive part of the formulas I lA, (A v B) and (B - A) and a negative part of the formulas lA, I(B I\A), (A - B).

We establish a connection between the semantics and syntax of classical sentential calculus by using the notions of "positive" and "negative parts". We define these notions purely syntactically and without reference to valuations and then prove they have the intended semantic properties.

We use certain nominal forms in order to give the syntactic definition of

1. Classical Sentential Calculus 11

positive and negative parts. By an n-place simple nominalform (n~ 1) we mean a finite string of symbols in which besides the primitive symbols of our formal language (sentential variables, connectives and round brackets) the nominal symbols *1' ... , *n occur just once. If $ is an n-place simple nominal form (n~ 1) and r1 , ••• , rn are non-empty finite strings of symbols then $[rl' ... , rn] denotes the result of replacing the nominal symbols *1' ... , *n in $ by r 1 , ••. , r n , respectively.

P-forms (positive forms) and N-forms (negative forms) are inductively defined as I-place simple nominal forms as follows:

PNFI. *1 is a P-form. PNF2. If 9 is a P-form and B is a formula, then 9[( * 1 vB)], 9[(B v * 1)] and

9[(B --4 * 1)] are P-forms and 9[, * 1] and 9[( * 1 --4 B)] are N-forms. PNF3. If % is an N-form and B is a formula, then %['*1] and %[(*1 --4..L)]

are P-forms and %[(*1 A B)] and %[(B A *1)] are N-forms. We have the following consequences of this inductive definition:

PNF4. No P-form is an N-form. PNF5. If 9 is a P-form and $ is a P-form (N-form), then 9[$] is a P-form

(N-form). PNF6. If $ is a P-form or an N-form and A is a formula, then $[A] is also a

formula.

Definition. A formula A is said to be a positive (negative) part of a formula F, if there is a P-form (N-form) $ such that $[A] is the formula F.

The rules PNFI-PNF3 yield the following inductive characterization of the positive and negative parts of a formula F.

PNI. Fis a positive part of F. PN2. If, A is a positive (negative) part of Fthen A is a negative (positive) part

of F. PN3. If(A v B) is a positive part of F, then A and Bare both positive parts of F. PN4. If(A A B) is a negative part of F, then A and Bare both negative parts of F. PN5. If (A --4 B) is a positive part of F, then A is a negative part and B is a

positive part of F. PN 6. If (A --4 ..L) is a negative part of F, then A is a positive part of F.

Example. If Fis the formula ,(,(Vi --4 V2) A (V2 v ..L)) then we getthepositive and negative parts of F as follows:

1. By PNI Fis a positive part of F. 2. By 1 and PN2 (, (Vi --4 V2) A (V2 v .1.)) is a negative part of F. 3. By 2 and PN4 ,(Vi --4 V2) and (V2 v.1.) are negative parts of F. 4. By 3 and PN2 (Vi --4 V2) is a positive part of F. 5. By 4 and PN5 Vi is a negative and V2 a positive part of F. These are all the positive and negative parts of F. By an NP-form we mean a 2-place simple nominal form f2 such that for every

formula B f2[ *1' B] is an N-form and f2[B, *1] is a P-form.

Examples. (*1--4*2), ('(*lAV)V(VV*2)) and «*2--4..L)--4(*l--4V» are NPforms.

12 I. Fundamentals

Convention. From now on &, &i will always denote P-forms, %; %i N-forms and !2, !2i NP-forms.

In the next few theorems we establish the semantic properties of positive and negative parts.

Theorem 1.5. If tff is a P-form (N-form) and Va sentential valuation with VA = t (VA=f), then Vtff[A]=t.

Proofby induction on the length of tff. 1. tffis the P-form *1' Then tff[A] is the formula A so by hypothesis Vtff[A]=t. 2. tff is a P-form &[( * 1 V B)], &[(B v * 1)] or &[(B - * 1)]. Now VA = t implies

V(A v B) = V(B v A) = V(B - A) = t. By I.H. (induction hypothesis) we have Vtff[A] = t.

3. tff is an N-form &[-'*1] or &[(*1 - B)]. Now VA=fimplies V-,A= V(A - B) = t. By I.H. we have V tff[A] = t.

4. tff is a P-form %[-'*1] or %[(*1-1.)]. Now VA=t implies V-,A= V(A - 1.) =! By I.H. we have V tff[A] = t.

5. tff is an N-form % [(*1 /\ B)] or % [(B /\ *dJ. Now VA = f implies V(A /\ B) = V(B /\ A) =! By I.H. we have Vtff[A] = t.

By the definition of P-form and N-form all the cases have been treated.

Definition. A positive part is said to be minimal if it is not of the form -, A, (A v B) or (A - B). A negative part is said to be minimal if it is not oftheform -,A, (A /\ B) or (A- 1.).

Theorem 1.6. If, under a sentential valuation V, every minimal positive part of a formula F takes the value f and every minimal ne.gative part of F takes the value t, then every positive part of F takes the value f and every negative part of F takes the value t. (In particular, VF= f, since F is a positive part of F.)

Proofby induction on the length of the positive (or negative) part C of the formula F. By hypothesis we need only treat the following cases which are those where Cis not a minimal positive or negative part.

1. C is a positive (negative) part -, A. Then A is a negative (positive) part of F. By I.H. VA=t (VA=f) hence VC=f(VC=t).

2. C is a positive part (A v B) (negative part (A /\ B». Then A and B are positive (negative) parts of F. By I.H. VA=B=f(VA= VB=t), hence VC=f(VC=t).

3. C is a positive part (A - B). Then A is a negative and B a positive part of F. By I.H. VA=t and VB=f, hence VC=!

4. C is a negative part (A - 1.). Then A is a positive part of F. By I.H. VA = f, hence VC=t.

6. Syntactic Characterization of Valid Formulas

A formula is said to be reducible if it is of the form &[(A /\ B)], %[(A v B)] or

2. Formal Systems 13

%[(A ~ B)] where, in the last case, B is not the formula . .1. Otherwise it is said to be irreducible.

Theorem 1.7. An irreducible formula is valid if, and only if, it is oftheform oP[v, v] or %[1..] (that is, if it contains a sentential variable v both as a positive and as a negative part or has 1.. as a ne.qative part).

Proof 1. Let V be an arbitrary sentential valuation. If Vv = t, then by Theorem 1.5 VoP[v,v]=t, since v occurs as a positive part in oP[v,v]. If Vv=J, then by Theorem 1.5 VoP[v, v] = t, since v occurs as a negative part in oP[v, v]. In either case, therefore, VoP[v, v]=t. By Theorem 1.5 V%[1..]=t, since V1..=! Hence oP[v, v] and %[ 1..] are valid formulas.

2. Let Fbe an irreducible formula which is not of the form oP[v, v] or %[1..]. We have to prove that Fis not valid. Since Fis irreducible every minimal positive part and every minimal negative part of F is either a sentential variable or the formula 1... Since F is not of the form %[ 1..] every minimal negative part of F is a sentential variable. Since Fis not ofthe form oP[v, v] no sentential variable occurs both as a positive and as a negative part of F. Let V be the valuation such that Vv = t if, and only if, the sentential variable v occurs as a negative part of F. Under this valuation Vevery minimal positive part of Ftakes the value f and every minimal negative parttakes the value t. By Theorem 1.6 we have VF=! Hence Fis not valid.

Theorem 1.8. The following hold for reducible formulas:

(1) &I[A 1\ B)] is valid if, and only if, &I[A] and &I[B] are valid. (2) %[(A v B)] is valid if, and only if, %[A] and %[B] are valid. (3) JV[(A ~ B)] is valid if, and only if, %[(A ~ 1..)] and %[B] are valid.

Proof of (I). I. Under a valuation Veither V(A 1\ B) = V A or V(A 1\ B) = VB, so V&I[(A 1\ B)] = V&I[A] or V&I[(A 1\ B)] = V&I[B]. Consequently, if &I[A] and &I[B] are valid, then &I[(A 1\ B)] is also valid.

2. Suppose &I[A] is not valid. Then there is a valuation V such that V&I[A] =! By Theorem 1.5 we have VA=! But then V(AI\B)=! Hence V&I[(AI\B)]= V&I[A] =! Therefore &I[(A 1\ B)] is not valid. Similarly, if &I[B] is not valid then &I[(A 1\ B)] is not valid.

The proof of (2) proceeds as for (1). (3) follows from (2) since %[(A ~ B)] is semantically equivalent to

%[( --, A v B)] and %[ ~ A] is semantically equivalent to %[(A ~ 1..)].

§2. Formal Systems

1. Fundamentals

The formal systems we are going to deal with are defined syntactically in four steps. 1: 1. Certain symbols (in general denumerably infinitely many) are introduced

as primitive symbols.

14 I. Fundamentals

~2. Certain finite strings of primitive symbols are selected as formulas by an inductive definition.

~3. Formulas of a particular sort (in general determined by formula schemata) are defined to be axioms.

~4. Certain configurations of the form

A 1,···,An I-B

where A 1 , .•. , An' B are formulas and in general n= I or n=2, are defined to be basic inferences. A formula on the left of the symbol I- is said to be a premise of the given basic inference. The formula on the right of the symbol I- is said to be the conclusion of the given basic inference.

Re ~l. We shall not in fact specify explicitly what all the primitive symbols are to be. We shall merely assume that the several primitive symbols are identifiable as such and that primitive symbols of different kinds are different from one another. (For example, a primitive symbol introduced as a connective shall not ~imultaneously be used as a propositional variable which is a primitive symbol.) In talking about primitive symbols we shall sometimes use the primitive symbols themselves (for example, connectives or round brackets) and sometimes use syntactic symbols (for example, v, V 1 , V 2 , ... for the propositional variables which are primitive symbols).

Re ~2. Often, as well as formulas some further kinds of finite strings of primitive symbols are inductively defined. These are the terms and expressions. In order to define and characterise formulas we frequently use nominal forms.

By an n-place nominal form (n ~ 1) in a formal system ~ we mean a non-empty finite string of symbols which contains, in addition to the primitive symbols of the formal system~, at most the nominal symbols *1' ... , *n. These nominal symbols are to be distinct from the primitive symbols of the formal system. In an n-place nominal form the nominal symbols *1' ... , *n may occur in one or more places but need not occur at all. We shall always denote nominal forms by script letters.

If d is an n-place nominal form (n ~ 1) and r 1, ... , r n are non-empty finite strings of symbols, then d[r1' ... , rn] denotes the string of symbols which results from the nominal form d when the nominal symbols *1' ... , *n are replaced at all occurrences in d by r1> ... , r n , respectively.

We have already defined some nominal forms, in §1 as P-forms, N-forms and NP-forms.

2. Deducible Formulas

Deducibility of formulas in a formal system is determined by the axioms and basic inferences according to the following inductive definition.

Dl. Every axiom is deducible. D2. If every premise of a basic inference is deducible, then the conclusion of

the inference is deducible. A proof (demonstration) of the deducibility of a formula F (according to D I

and D2) is said to be a deduction of F.


Sometimes we need to specify the deducibility of a formula more closely so to the formula we assign a natural number, which serves as the order, by the following inductive definition:

DO 1. Every axiom is deducible with order O. DO 2. If the premises A 1 , ••• , An (n~ 1) ofa basic inference are deducible with

orders m l , ..• , mn , respectively, then the conclusion of the inference is deducible with order max (m 1 , ••• , mn ) + 1.

A formula may be deducible with many different orders. However, given that a formula is deducible we may always assume that it is deducible with some fixed order.

We write I-F to indicate that the formula F is deducible. We prove theorems about deducible formulas by induction on the deduction.

This induction goes as follows. First we prove that the theorem to be proved holds for every axiom. The second

step is to prove the theorem holds for every conclusion of a basic inference under the induction hypothesis that the theorem holds for every premise ofthe given basic inference. When both these steps have been completed then it follows from the above inductive definition of deducibility that the theorem has been completely proved.

A formal system is said to be consistent if not every formula of the system is deducible. Consistency is the least condition one can place on a formal system if deducibility is to be meaningful since otherwise being deducible adds nothing to tqe notion of formula.

3. Permissible Inferences

A configuration

A 1 , ••. ,An I-B

where A 1, .•• , An and B are formulas (n ~ 1) is said to be a permissible inference if the following holds: if A l, ... , An are deducible then B is also deducible.

In particular every basic inference is a permissible inference. In the formal systems we treat in the sequel we consider many more inferences which, though not defined to be basic inferences, will be shown to be permissible. These ,will be used to establish completeness of the notion of deducibility.

A weak inference is a permissible inference A I- B such that: if the formula A is deducible with order m, then the formula B is deducible with order ~m.

An inference

may be permissible for several different reasons. It may happen that B is deducible from the formulas A l , ... , An by using the

axioms and further basic inferences. In this case we say that the given inference is

16 I. Fundamentals

directly derivable. Clearly a directly derivable inference remains permissible if we extend the notion of deducibility by adding axioms or basic inferences to the formal system.

But it may also happen that the inference can only be proved permissible by referring to the assumed deductions of the formulae A 1 , ••• , An. Such an inference, which is not directly derivable, may cease to be permissible if deducibility in the formal system is extended. In what follows we shall principally be concerned with permissible inferences of this second kind.

Example. Let the formulas of a formal system L be those of the sentential calculus defined in § 1.4. Let the only axioms of the system L be those formulas of the form

(A viA).

Let the only basic inferences of the system L be those inferences of the forms

A ~ (A v B) and B~ (A v B).

Obviously

B~((AvB)vC)

is a directly derivable inference of the system L since one can deduce (A v B) from B by a basic inference and then deduce ((A v B) v C) from (A v B).

Every inference of the form

(A v A)~A

is permissible in L since a formula (A v A) can only be deduced from the basic inference A ~ (A v A). So the formula (A v A) is only deducible if the formula A is deducible. Now if we extend the system L by adding all axioms of the form (A v A), then it is no longer the case that every inference (A v A) ~ A is permissible since then, for example, the formula (..1 v ..i) is deducible while the formula ..1 is not deducible in the extended system. Thus the inference (A v A) ~ A is permissible in L but not directly derivable.

4. Sentential Properties of Formal Systems

A formal system is said to be sententially closed (s-closed for short) if among its primitive symbols it contains the connectives from a complete set of connectives which are used for joining formulae as in the sentential calculus. All sentential connectives can be defined in such a formal system because of the completeness of the underlying system of connectives.

A formula in a formal system is said to be s-valid if it is obtained by substituting formulas of the formal system for the sentential variables in a valid formula A of the


classical sentential calculus where every senteniial variable which occurs in more than one place in A is always replaced by the same formula of the formal system.

Example. If~ is a formal systemcontaining~ as a primitive symbol and (A ~ A) is a formula of the system ~, then this formula is s-valid.

A formal system is said to be s-complete if it is s-closed and every s-valid formula of the system is deducible in the system.

Note that the notions of s-closure, s-validity and s-completeness defined above refer to the classical sentential calculus and not to the intuitionistic sentential calculus which forms a part of the intuitionistic predicate calculus we treat in chapter III.

A formal system in which every formula A has a negation -, A defined is said to be s-consistent if there is no formula A such that A and -, A are both deducible in the system. Every s-consistent system is obviously consistent. In many formal systems the converse is easily established, that consistency implies s-consistency. We shall only deal with formal systems which are s-consistent.

5. The Formal System CS of the Classical Sentential Calculus

Primitive symbols of the system CS: Denumerably infinitely many sentential variables, the conneCtives 1.., -', A, v, ~ and round brackets.

• Theformulas of the system CS are defined as in §1.4 and P-forms, N-forms and NP-forms as in §l.5.

As in §1 we use the following syntactic symbols: v for sentential variables, A, B, F for formulas, fJJ for P-forms, % for N-forms and .?l for NP-forms.

The axioms of the system CS are all formulas of the forms:

(Ax. I) .?l[v, v] (Ax. II) %[1..]

The basic inferences of the system CS are all inferences of forms:

(fA) fJJ[A],fJJ[B]~fJJ[(AAB)] (I v) %[A], %[B] ~ %[(A v B)] (I ~) %[(A ~ 1..)], %[B] ~ %[(A ~ B»).

The next two theorems show that (syntactic) deducibility given by the above coincides with the semantic notion of validity.

Consistency Theorem. Every deducible formula is valid.

Proof by induction on the deduction. It follows from Theorem 1.5, that every axiom is valid. By Theorem 1.8, if both premises of a basic inference are valid then so is the conclusion. Hence every deducible formula is valid.

18 I. Fundamentals

Completeness Theorem. Every validformula is deducible.

Proof Let Fbe a valid formula. Define the reducibility degree of Fto be the number of connectives A, v and --4 occurring in the minimal positive and minimal negative parts of F. We prove by induction on the reducibility degree of Fthat the formula F is deducible.

1. F is an irreducible formula. Then by Theorem 1. 7 F is an axiom and hence is deducible.

2. F is a reducible formula. Then F is of the form of the conclusion of a basic inference both of whose premises have smaller reducibility degree than F and hence by Theorem 1.8 are valid. By the induction hypothesis these premises are therefore deducible. But then F is deducible by means of the basic inference involved.

It follows from the consistency theorem and the completeness theorem that the system CS is s-consistent and s-complete.

Remark. For every formula of the system CS we can decide in an elementary way whether and how the formula is deducible in the system CS. For we can easily see whether a formula has the form of an axiom or of a conclusion of a basic inference and because every premise of a basic inference has a lower reducibility degree than the conclusion of the inference.

Chapter II

Classical Predicate Calculus

Predicate calculus forms an extension of the sentential calculus in two different regards.

1. As well as variables for arbitrary sentences we use variables ranging over an arbitrary (non-empty) object domain and arbitrary one- and many-place predicates over the object domain. So in addition to sentential variables we have as basic formulas, formulas of the form p(al' ... , an) with the interpretation: "The objects ai' ... , an satisfy the n-place predicate p".

2. As well as the connectives 1\ ("and"), v ("or") ... of sentential calculus we use the quantifiers 'i and 3 which can be employed to form formulas ofthe following kinds:

. 'ix$'[x] interpreted "For all objects x, $'[x] holds". 3x$'[x] interpreted "There is an object x, such that $'[x] holds". We shall make the interpretation of these formulas more precise in §5 by an

appropriate semantics but first we develop a pure syntax for classical predicate calculus. For this we shall use only the connectives .1 and ~ and only the universal quantifier 'i as primitive symbols since the remaining connectives and the existential quantifier 3 can be defined from these in classical predicate calculus. We could in fact use instead the connectives of any other complete system and 3 as primitive symbols. The choice of .1, ~ and 'i as primitive symbols turns out to be especially suitable for other parts oflogic as well, in particular for intuitionistic type theory.

§3. The Formal System CP

1. Primitive Symbols of the system CP: 1.1. Countably infinitely many free and bound object variables, sentential

variables and predicate symbols of each number of arguments ~ 1. 1.2. The logical symbols .1, ~ and'i. 1.3. Round brackets and the comma. As syntactic variables we use a, ai for free object variables, x, Xi for bound object variables, v, Vi for sentential variables.

20 II. Classical Predicate Calculus

2. Inductive Definition of the formulas of the system CP. 2.1. Every sentential variable is a formula. 2.2. The symbol ..1 is aformula. 2.3. If P is an n-place predicate symbol (n~ 1) and al' ... , an are free object

variables then p(a1, . .. , an) is aformula. 2.4. If A and B are formulas, then (A - B) is also aformula. 2.5. If ff[a] is aformula and x a bound object variable which does not occur

in the I-place nominal form ff, then \ixff[x] is also aformula. The formulas defined by 2.1-2.3 are said to be atomic (or prime) formulas. As syntactic symbols we use A, Ai' B, Bi, C, C i , F, Fi , G, G i for formulas, P, Pi

for atomic formulas, ff, ffi for I-place nominal forms such that ff[a] , ff;[a] are formulas.

3. P-forms and N-forms are defined as in §1.5. In the system CS since the only connectives which occur as primitive symbols are ..1 and - we have just the following rules for the inductive definition:

3.1. *1 is a P-form. 3.2 If ?J> is a P-form and B a formula; then .9[(B - * d] is a P-form and

?J>[(*1- B)] is an N-form. 3.3. If % is an N-form, then %[(*1 - ..i)] is a P-form. 4. For positive and negative parts of a formula F, which are determined by

P-forms and N -forms as in § 1.5, we have the following inductive characterization: 4.1. F is a positive part of F. 4.2. If (A - B) is a positive part of F, then A is a negative and B a positive part

ofF. 4.3. If (A - ..i) is a negative part of F, then A is a positive part of F. As in §l we use syntactic symbols as follows: ?J>, .9i for P-forms, %, %i for

N-forms and fl, fli for NP-forms. A positive part ofa formula is said to be minimal if it is not of the form (A - B).

A negative part is said to be minimal if it is not of the form (A - ..i). F ~ G ("G follows structurally from F") denotes that every minimal positive

part of F also occurs as a positive part of G and every minimal negative part of F also occurs as a negative part of G.

5. The axioms of the system CP are all formulas of the form

(Ax. I) fl[P, P] (Ax. II) %[..1].

6. The basic inferences of the system CP are all inferences of the forms

(Sl) %[(A - ..i)], %[B] f- %[(A - B)] (S2) .9[ff[a]] f- ?J>[\ixff[x]] with a condition on the variable a (S3) (\ixff[x] - %[ff[a]]) f- %[\ixff[x]].

The free object variable denoted by a in the premise ofa basic inference (S2) is said to be the eigenvariable of the inference. The condition on the variable of an (S2) inference is: The eigenvariable must not occur in the conclusion.

4. Deducible Formulas and Permissible Inferences 21

The indicated positive or negative part in the conclusion of a basic inference is said to be the principal part of the given basic infe;ence. In an (S I)-inference we may assume that its principal part is not of the form (A --> ..i) since otherwise the conclusion would be the same as the first premise. Hence every principal part of a basic inference is a minimal positive or negative part of the conclusion.

No atomic formula is deducible in CP since an atomic formula cannot occur as an axiom nor as the conclusion of a basic inference. Hence the system CP is consistent. We shall see in §4 that it is also s-complete and s-consistent.

Remark. The notions of "negative part'" and "positive part" may be regarded as generalizations of G. Gentzen's "antecedent formula" and "succedent formula". The generalizations turn out so that there is no need to have any structural inferences as basic inferences. In §4 we shall prove that the inferences corresponding to Gentzen's structural inferences are permissible. Then Theorem 4.6 corresponds to Gentzen's Hauptsatz on the elimination of the cut rule.

§4. Deducible Formulas and Permissible Inferences

. 1. Generalizations of the Axioms

Theorem 4.1. Every formula of the form ~[C, C] is deducible.

Proofby induction on the number of --> and V symbols occurring in C. 1. C is an atomic formula. Then ~[C, C] is an axiom. 2. C is a formula (A --> B). Since ~ is an NP-form so are ~[( *2 --> ..i), (* 1 --> B)]

and ~[*1' (A --> *2)]. By I.H. (induction hypothesis) both of the formulas ~[(A --> ..i), (A --> B)] and ~[B, (A --> B)] are deducible. Hence ~[(A --> B), (A --> B)] is deducible by an SI-inference.

3. C is a formula Vxff[x]. We choose a free variable a which does not occur in ~[C, C]. Since ~ is an NP-form so is (C --> ~). By I.H. the formula (Vxff[x]--> ~[ff[a], ff[a]]) is therefore deducible. By an (S3)-inference we have ~[Vxff[x], ff[a]] and then by an (S2)-inference we have ~[Vxff[x], Vxff[x]].

Corollary to Theorem 4.1. The system CP is s-complete.

Proof By Corollary 1.4 ..1 and --> form a complete set of connectives. By §2.5 it follows that every s-valid formula which contains no connectives other than ..1 and --> is deducible from formulas only of the forms ~[C, C] and fi[..1] using (S 1)inferences. By Theorem 4.1 it follows that CP is s-complete.


2. Weak Inferences

Theorem 4.2 (Substitution rule). If a 1 does not occur in the nominalform ff, then

ff[a 1] f- ff[a z]

is a weak inference.

Proofby induction on the deduction. Suppose the formula ff[a 1] is deduced with order m.

1. Suppose ff[a 1] is an axiom. Then ff[a z] is also an axiom. 2. Suppose ff[a 1] has been inferred by an (Sl) or (S3) basic inference from

Fi (i= 1, 2 or i= 1). Fi is a formula ~[a1] where a1 does not occur in ffi . By the induction hypothesis ffi[a Z ] is deducible with order <m.

~[az] (i= 1, 2 or i= 1) f- ff[az] is then an (SI) or (S3) basic inference. Hence ff[a z] is deducible with order ~ m.

3. Suppose ff[a 1] has been inferred from F1 by an (S2)-inference. We choose a free object variable ao, different from a1 and az which does not occur in ff. Let Fo be the formula obtained from F1 by replacing the eigenvariable of the (S2)inference, wherever it occurs in F1, by ao. By I.H. Fo is deducible with order <m. Fo is a formula ff 0[a 1] where at does not occur in ff o. By I.H. ff o[az] is also deducible with order <m. Now

is an (S2)-inference with eigenvariable ao. Hence ff[az] is deducible with order ~m.

Theorem 4.3 (Inversion rules). The following are weak inferences

(1) .H[(A --> B)] f- .H[(A --> 1.)] (2) .H[(A --> B)] f- .H[B] (3) 9'[\fxff[x]] f- &>[ff[a]].

1. Proofof(1) and (2) by induction on the deduction. Suppose .H[(A --> B)] is deduced with order m. If B is the formula 1. thyn the assertions are trivial since then the conclusion of(1) is the same as the premise and the conclusion of(2) is an axiom. Now let B be a formula different from 1..

1.1. Suppose .H[(A --> B)] is an axiom. Then .H[(A --> 1.)] and .H[B] are axioms.

1.2. Suppose .H[(A --> B)] has been inferred by an (SI)-inference whose principal negative part is the part «A --> B) shown. Then the premises .H[(A --> 1-)] and .H[B] are deducible with order <m.

1.3. Suppose .H[(A --> B)] has been deduced in some other way by a basic inference from Fi (i= 1, 2 or i= 1). Then Fi is a formula JV;[(A --> B)] such that

JV;[(A --> 1-)] (i= 1,2 or i= 1) f- .H(A --> 1-)


and

%;[BJ (i= 1,'2 or i= 1) f- %[BJ

are basic inferences. Hence the assertion holds by the induction hypothesis.

2. Proof of (3) by induction on the deduction. Suppose &>['v'xff[xJJ is deduced with order m.

2.1. Suppose &>[\fxff[xJJ is an axiom. Then 2Ji[F[aJJ is also an axiom. 2.2. Suppose &>[\fxff[xJJ has been inferred by an (S2)-inference whose

principal part is the given positive part \fxff[x]. The premise &>[ff[aoJJ of this basic inference is deducible with order m - 1 where ao is the eigenvariable of the (S2)inference which does not occur in the nominal form 2Ji[ff]. It follows from Theorem 4.2 that &>[ff[aJ] is deducible with order <m.

2.3. Suppose 2Ji[\fxff[xJJ has been deduced in some other way by a basic inference. We choose a free object variable a l which is not the eigenvariable of the basic inference and which does not occur in the nominal form &>[ ff]. As in case 1.3 it then follows from I.H. that the formula 2Ji[ff[a l JJ is deducible with order ~m. By Theorem 4.2 &>[ff[aJJ is also deducible with order ~m.

Theorem 4.4 (Structural rule). ifF fi G, then F f- G is a weak inference.

P.roof by induction on the deduction. Suppose the formula F is deducible with order m.

1. Suppose F is an axiom. Then it follows from F fi G that G is also an axiom. 2. Suppose F is a formula % l[(A ---+ B)J which has been inferred from

% 1 [( A ---+ 1-) J and % 1 [BJ by an (S 1 )-inference. We replace every negative part (A ---+ B) in % l[(A ---+ 1-)J by (A ---+ 1-) and every negative part (A ---+ B) in % l[BJ by B. Then we obtain formulas Fl and F2 which by Theorem 4.3 are deducible with orders <m. It follows from Ffi G that G is a formula.%2[(A ---+ B)J and that Fl fi % 2[(A ---+ 1-)] and F2 fi % 2[B]. By I.H. the formulas % 2[(A ---+ 1-)] and %2[BJ are therefore deducible with orders <m. It follows by an (SI)-inference that % 2[(A ---+ B)J is deducible with order ~m.

3. Suppose Fis a formula 2Ji l [\fxff[xJJ inferred from 2Ji l [ff[aJJ by an (S2)inference. We choose a free object variable ao which occurs neither in Fnor in G. In the formula 2Ji l [ff[aJJ we replace a by ao and every positive part \fxff[xJ by ff[ao]. This gives a formula Fo which, by Theorems 4.2 and 4.3, is deducible with order <m. It follows from Ffi G that G is a formula &>2[\fxff[xJJ and Fo fi 2Ji2[ff[aoJJ holds. By I.H. the formula &>2[ff[aoJJ is deducible with order <m. It follows by an (S2)-inference that 2Ji2[\fxff[xJJ is deducible with order ~ m.

4. Suppose F is a formula % 1 [\fxff[xJJ inferred from (\fxff[xJ ---+ % 1 [ff[aJJ) by an (S3)-inference. It follows from Ffi G that G is a formula % 2[\fxff[xJJ and (\fxff[xJ ---+ % 1 [ff[aJJ) fi (\fxff[xJ ---+ % 2[ff[aJJ) holds. By I.H. the formula (\fxff[xJ ---+ % 2[ff[aJJ) is therefore deducible with order <m. It follows by an (S3)-inference that %2[\fxff[xJJ is deducible with order ~m.


3. Further Permissible Inferences

Theorem 4.5. JV[ff[a]] ~ JV[\fxff[x]] is a permissible inference (provided x does not occur in ff).

Proof We have JV[ff[a]] f! (\fxff[x] -> JV[ff[a]]). From ~ JV[ff[a]] by Theorem 4.4 we therefore have ~ (\fxff[x] -> JV[ff[a]]). By an (S3)-inference ~ JV[\fxff[x]] follows.

Theorem 4.6 (Cut rule).

&[A], (A -> B) ~ &[B]

is a permissible inference.

Proofby induction on the number of -> and \f symbols occurring in the formula A. 1. Suppose A is an atomic formula. For this case we have to do a subsidiary

induction on the order of a deduction of the formula .9'[A]. 1.1. Suppose .9'[A] is an axiom. Then either &[B] is also an axiom or.9' is a

P-form 2[A, *1]. In the latter case (A -> B) f! &[B]. Then by Theorem 4.4 from ~ (A -> B) we have ~ .9'[B].

1.2. Suppose &[A] was obtained by a basic inference from Fi (i = 1,2 or i = I). Then by Theorem 4.2 we may assume that the basic inference has no free object variable which occurs in B as its eigenvariable. Then Fi is a formula of the form &;[A] so

.9'i[B] (i= I, 2 or i= 1) ~.9'[B]

is also a basic inference. Then ~~[B] follows from ~~[A] and ~(A -> B) by the S.I.H. (subsidiary induction hypothesis). Using the given basic inference ~.9'[B] follows.

2. Suppose A is a formula Al -> A2. By Theorem 4.3 from H(A l -> A2) -> B) we have H(A l -> 1..) -> B) and ~(A2 -> B). Since.9' is a P-form so is .9'[(Al -> *dJ. Hence by the I.H. from ~&[(Al -> A2)] and ~(A2 -> B) we have ~.9'[Al -> B)]. By Theorem 4.4 there follows ~(Al -> .9'[B]). Since «*1 -> 1..) -> B) is a P-form, from ~«Al -> 1..) -> B) and ~(Al -> &[B]) by I.H. we have ~«.9'[B] -> 1..) -> B). We have «.9'[B] -> 1..) -> B) f! .9'[B]. Therefore by Theorem 4.4 we have ~&[BJ.

3. Suppose A is a formula \fxff[x]. In this case we use a subsidiary induction on the order m of a deduction of the formula (\fxff[x] -> B).

3.1. Suppose (\fxff[x] -> B) is an axiom. Then .9'[B] is also an axiom. 3.2. Suppose (\fxff[x] -> B) was inferred by an (SI) or (S2) basic inference

from Fi (i= I, 2 or i= I). Then by Theorem 4.2 we may assume that the basic inference has no free object variable which occurs in .9' as its eigenvariable. Then Fi is a formula of the form (\fxff[x] -> Bi ) so

.9'[BJ (i= I, 2 or i= I) ~.9'[B]


is also an (Sl) or (S2) basic inference. By the S.I.H. from f-~[Vxff[x]] and f-(Vxff[x] --> Bi ) we have f-~[BJ. Then by the given basic inference we have f-g>[B].

3.3. Suppose (Vxff[x] --> B) was inferred by an (S3)-inference from (Vxff[x] --> (ff[a] --> B). By the S.I.H. we then have f- g>[(ff[a] --> B)]. f-(ff[a]--> g>[B]) then follows by Theorem 4.4. By Theorem 4.3 from f- ~[Vxff[x]] we have f-g>[ff[a]]. Then using f-(ff[a] --> ~[B]) we have by the I.H. f-&[~[B]]. Now g>[2P[B]] ~ &fB]. Therefore by Theorem 4.4 we have f- g>[B].

3.4. Suppose Bis a formula JV[Vxoffo[xo]] and (Vxff[x] --> JV[Vxoffo[xo]]) was inferred from (Vxoff o[xo] --> (Vxff[x] --> JV[ff ora]])) by an (S3)-inference. By Theorem 4.4 (Vxff[x] --> (Vxoffo[xo] --> JV[ffo[a]])) is also deducible with order <m. Hence by the S.I.H. we have f-g>[(Vxoffo[xo] --> JV[ffo[a]])]. By Theorem 4.4 we therefore have f-(V-xoffo[xo] --> g>[JV[ffo[a]]]). Since g>[JV] is also an N-form, using an (S3)-inference we have f- g>[[JV[Vxoff o[xo]]].

4. Defined Logical Connectives

The connectives -', /\, v and the existential quantifier 3 are defined as follows in the system CP :

-,A:=(A--> 1-) (A /\ B): =«A --> (B--> 1-)) --> 1-) (A v B) : = «A --> 1-) --> B)

3xff[x] :=(Vx(ff[x]--> 1-)--> 1-).

These connectives we have just defined satisfy the defining rules given in § 1. 5 for positive and negative parts as we now show.

If -, A (that is (A --> 1-)) is a positive (or negative) part of a formula F, then A is a negative (or positive) part of F.

If«A --> (B --> 1-)) --> 1-) is a negative part ofF, then (A --> (B--> 1-)) is a positive part, A is a negative part, (B --> 1-) a positive part and B a negative part of F. So we have: If (A /\ B) is a negative part of F, then A and B are also negative parts of F.

If «A --> 1-) --> B) is a positive part of F, then (A -+ 1-) is a negative part, A a positive part and B a positive part of F.

So we have: If (A v B) is a positive part of F, then A and B are also positive parts of F.

As we have defined negation, we therefore have:

Theorem 4.7. The formal system CP is s-consistent.

Proof By Theorem 4.6 and the definition of -,

A, -,A f- 1-

is a permissible inference. Since the formula 1- is not deducible in CP it therefore follows that there is no formula A such that A and -,A are both deducible in CPo


The proof-theoretic properties of A, v and 3 are given by the following theorem.

Theorem 4.8. The following inferences are permissible:

(1) &P[A], &P[B] f- &![(A A B)] (2) ..;V[AJ f- ..;V[(A1 AAz)] (i= 1, 2) (3) &P[AJ f- &P[(A1 v Az)] (i= 1, 2) (4) ";v[A], ..;V[B] f- ..;V[(A v B)] (5) &P[~[a]] f- &P[3x~[x]] (6) ..;V[~[a]] f- ";v[3x~[x]],

if a does not occur in the conclusion (condition on variables).

Proof (1) For every formula Cwe have &P[C] ~ &P[«C - ..l) - ..l)]. TherefQre by Theorem 4.4 from f-&P[A] and f- &P[B] we have f- &P[«A - ..l) - ..l)] and f-&P[«B- ..l)- ..l)]. Since &P[(*1 - ..l)] is an N-form, we obtain by an (SI)inference f-&P[«A - (B - ..l» - ..l)], that is f- &P[(A A B)].

(2) Follows from Theorem 4.4 since ..;V[AJ ~ ..;V [(A 1 AAz)] (i= 1, 2) by the definition of A .

(3) Follows from Theorem 4.4 since &P[AJ ~ &P[(A1 v Az)] (i= 1, 2) by the definition of v .

(4) We have ";v[A] ~ ";v[«A - ..l) - ..l)]. By Theorem 4.4 from f- ..;V [A] we therefore have f- ..;V [«A - ..l) - ..l)]. Using f- ..;V[B] by an (SI)-inference we have f-..;V[«A - ..l) - B)], that is ..;V[(A v B)].

(5) By Theorem 4.4 from f-&P[~[a]] we have f-&P[«~[a] - ..l) - ..l)]. Since &P[(*1 - ..l)] is an N-form Theorem 4.5 yields f-&P[(\fx(~[x] - ..l) - ..l)], that is &P[3x~[x]].

(6) From f-..;V[~[a]] by Theorem 4.4 we have f-..;V[«~[a] - ..l) - ..l)]. ..;V[( * 1 - ..l)] is a P-form. If a does not occur in ..;V[~], then by an (S2)-inference we have f-";v[(\fx(~[x] - ..l) - ..l)], that is f-";v[3x~[x]].

Remark. All the proofs we have given in this section which show that certain inferences are permissible are constructive. That is to say: In each case, if deductions of the formulas A l' ... , An are given, then one can obtain a deduction of the formula B from the proof of the permissibility of the inference

§5. Semantics of Classical Predicate Calculus

Although all the notions we have used so far and all the proofs we have given were strictly constructive (jinitist in the sense of D. Hilbert), when it comes to classical predicate calculus we have to deal with non-constructive notions.

5. Semantics of Classical Predicate Calculus 27

1. Classical Models

A model J( = <~, V) consists of a non-empty set ~ (the domain of individuals of the model) and a function V (the valuation of the model) defined on the free object variables, predicate letters and propositional variables as follows:

VI. For each free object variable a, Va is an element ofthe set~. V2. For each n-place predicate letter p (n~ 1), Vp is a set ofn-tuples «(Xl, .•• , (Xn)

of elements (Xl' .•• , (Xn from the set ~. V3. For each sentential variable v, Vv is one of the truth-values t,f By a ~-formula we mean a string of symbols which is the result of replacing

some free object variables in a formula of the system CP by names of elements from the set~.

We use the following as syntactic symbols: (X, (Xi for names of elements of the set ~, e, ei for free object variables and names of elements of the set~, A', B', F', \fx~'[x] for ~-formulas. For a model J(=<~, V) the valuation V has properties VI-V3. We now

extend Vby definition so that every ~-formula F' also gets a truth-value VF'. V4. If (X E~, then V(X: =(X.

VS. v.l.:=/ V6. If p is an n-place predicate letter (n~l) then Vp(el, ... ,en):=t if

(Vel, ... , Ven) E Vp. Otherwise, Vp(e l , ... , en)=/ V7. If VA'=for VB'=t, then V(A' -4 B') :=t. Otherwise V(A' -4 B') :=/ V8. If V~'[(X] = t for all (X E ~ then V\fx~'[x]: = t. Otherwise V\fx~'[x]: = /

We note in particular that every formula F gets a truth-value VF, since formulas are just particular ~-formulas.

VF' is defined by induction on the number of -4 and \f symbols which occur in the ~-formula F'. But this is non-constructive since we cannot always calculate the value VF' as the following example shows.

Example. Let ~ be the set of positive integers. Let p be a 4-place predicate letter and set Vp to be the set of quadruples «(Xl> •.• , (X4) of positive integers such that ~4+2 +~4+2 #(X~4+2. Let Fbe the formula \fx l \fX2 \fX3 \fX4P(x l , x 2 , x 3, x 4 ). Then in the model <~, V) we have VF=t if, and only if, Fermat's last theorem is true, that is iffor every n ~ 3 there are no positive integers (Xl' (X2' (X3 such that (X~ + (Xi = (X3·

But Fermat's conjecture has not yet been fully established. In the definition of VF' for a ~-formula F' the connectives..l and -4 are treated

as they were in § I in the definition of V A for a formula A ofthe sentential calculus. Consequently, corresponding to Theorem 1.5 we have: PN-Lemma. If cff is a P-form (or an N-form) and VA =t (or VA= f) for a valuation V in a model, then Vcff[A] = t.

A formula F is said to be true in a model J( = <~, V) if VF= t. We also say in this case that J( is a model of the formula F.

A formula is said to be valid if it is true in every model. It is said to be satisfiable if there is some model in which it is true.


2. The Consistency Theorem

Theorem 5.1 (Consistency Theorem). Every deducible formula is valid.

Proofby induction on the deduction. Let Fbe a deducible formula and .It = <!!J, V) an arbitrary model. We have to prove that VF=t.

1. SupposeFis an axiom ~[P, P] or ';v[1-]. By the PN-Iemma we have VF=t. 2. Suppose Fis a formula .;V[(A - B)] inferred from .;V[(A - 1-)] and .;V[B]

by an (S1)-inference. By I.H. we have V';v[(A - 1-)] = t and V.;V[B] = t. If VB= t, then V(A - B)= VB and V';v[(A - B)] = V';v[B]=t. If VB=j, then V(A - B)= V(A - 1-) and V';v[(A - B)] = V';v[(A - 1-)] = t. So in either case V';v[(A - B)] = t.

3. Suppose F is a formula &I[\fxff[x]] inferred from ,o//[ff[aJ] by an (S2)inference. If V\fxff[x] = t, then V&I[\fxff[x]] = t by the PN-Iemma. Now suppose there exists IX E!!J with Vff[lX] = f We define a new model .It' = <!!J, V') )Vhere V' a = IX and otherwise V' agrees with Von free object variables, predicate letters and sentential variables. Then, since the eigenvariable a does not occur in &I[ff] we obtain

and

V'ff[a] = Vff[IX]=f V'\fxff[x] = V\fxff[x] =f

V&I[\fxff[x]] = V' &I[\fxff[x]] = V' &I[ff[a]].

Since the premise &I[ff[a]] of the (S2)-inference is valid by I.H., we have V'&I[ff[a]]=t. Hence V&I[\fxff[x]]=t.

4. Suppose Fis a formula ';v[\fxff[x]] inferred from (\fxff[x] - .;V(ff[a]]) by an (S3)-inference. If VV'xff[x] = j, then V';v[\fxff[x]] = t by the PN-Iemma. Now suppose V\fxff[x] = t. Then Vff[a] = t too and it follows that V';v[\fxff[x]] = V.;V[ff[a]] = V(\fxff[x] - ';v[ff[a]]). By I.H. V(\fxff[x].;V [ff[a]]) = t. Hence we also have V.;V[\fxff[x]] = t.

3. The Completeness Theorem

In order to prove that, conversely, every valid formula is deducible we need several auxiliary notions.

First we let

be an enumeration of all the free object variables. A part of a formula F is said to be reducible if it is a positive part of the form

\fxff[x] or a negative part of the form \fxff[x] or (A - B) provided B is not the


formula ..l. A formula is said to be reducible if it has at least one reducible part, otherwise it is said to be irreducible. According to this definition a formula F is irreducible if, and only if, every minimal positive part and every minimal negative part of F is an atomic formula. The reducibility degree of a formula F is the number of -> and V symbols occurring in the reducible parts of F.

By the distinguished part of a reducible formula Fwe mean that reducible part of F which occurs furthest to the right.

AD-chain (deduction chain) for a formula F is a sequence of formulas

constructed as follows: 01. The initial formula Fo of the O-chain is the formula F. 02. If the formula Fn of the O-chain is an axiom or an irreducible formula,

then it is the last formula of the O-chain. We then say that the O-chain has length n. 03. If the formula Fn of the O-chain is a reducible formula which is not an

axiom then Fn has an immediate successor Fn + 1 in the O-chain determined as follows by Fo, ... , Fn.

03.1. If Fn is a formula %[(A -> B)] with distinguished negative part (A -> B), then Fn+ 1 is either the formula %[(A -> ..l)] or the formula %[B].

03.2. If Fn is a formula &>[Vx$'[x]] with distinguished positive part Vx$'[x], then Fn+ 1 is the formula &>[$'[IJ(J] where i is the least number such that lJ(i does not occur in Fn.

03.3. If Fn isa formula %[Vx$'[x]] with distinguished negative part Vx$'[x], then Fn+ 1 is the formula (Vx$'[x] -> %[$'[IJ(J]), where i is the least number such that $'[IJ(J does not occur as a negative part in any of the formulas among Fo, ... , Fn if the nominal symbol *1 occurs in $'.

O-chains are therefore formed inversely to the basic inferences. We shall shortly prove:

Principal Syntactic Lemma. If every D-chain of a formula F contains an axiom then the formula F is deducible.

Principal Semantic Lemma. If there is a D-chain of a formula F which contains no axiom, then there is a model <N, V) with domain the natural numbers N such that VF=f

From these two lemmata we obtain:

Theorem 5.2 (Completeness Theorem). Every valid formula is deducible.

Proof Let F be a valid formula. By the principal semantic lemma every O-chain of F contains an axiom. From the principal syntactic lemma it follows that F is deducible.

Theorem 5.3 (LOwenheim-Skolem). Every satisfiable formula has a model whose domain is the natural numbers.


Proof Let Fbe a satisfiable formula. Then the formula (F -> 1..) is not valid, so by Theorem 5.1, not deducible. From the principal syntactic lemma it follows that there is a D-chain for (F -> 1..) which contains no axiom. By the principal semantic lemma it follows that there is a model.A =(N, V) such that V(F -> 1..) =f Then VF= t so F is true in the model.A.

In order to prove the principal syntactic lemma we need the following version of Konig's Lemma:

If a formula F has infinitely many D-chains then there is an infinite D-chain for F.

Proof of the lemma. An infinite D-chain

is obtained from F as follows. Let Fa be the formula F. By hypothesis ther.e are infinitely many D-chains beginning with Fa. Now assume Fa, ... , Fn have been determined in such a way that infinitely many D-chains begin with Fa, ... , Fn. Then Fn is a reducible formula which is not an axiom.

1. Suppose Fn is a formula %[(A -> B)] with distinguished negative part (A -> B). Then only one of the two formulas %[(A -> 1..)] and %[B] can occur in D-chain after Fn. Let Fn+ 1 be the formula %[(A -> 1..)] if infinitely many D-chains begin with Fa, ... , Fn, %[(A -> 1..)]. Otherwise let Fn+l be the formula %[B].

2. Suppose Fn has a distinguished part of the form 'v'xff[x]. Then the formula Fn+ 1 following Fn is uniquely determined by Fa,· .. , Fn.

It follows from our assumption in each case that there are infinitely many D-chains which begin with Fa, ... , Fn+1 • Since this process of going from Fn to Fn+ 1 never stops, it determines an infinite D-chain of F. (The proof is nonconstructive since the choice of Fn+ 1 depends on infinitely many successors.)

Proof of the principal syntactic lemma. Let F be a formula such that every Dchain of F contains an axiom. Then such an axiom is the last formula of such a D-chain. Hence every D-chain of F is finite. From Konig's lemma it follows that there are only finitely many D-chains. So these have a maximal length m. We prove by induction on m-n that every formula Fn which occurs as the n-th formula in a D-chain of F is deducible. If Fn is the last formula of a D-chain of F then Fn is an axiom. Otherwise Fn is of the form of a conclusion of a basic inference whose premises come after Fn in D-chains of F. By I.H. these premises are deducible. Consequently Fn is also deducible. In particular the formula Fa, that is, F, is deducible, as was to be shown.

Proof of the principal semantic lemma. Let

be a D-chain which contains no axiom where Fa is the formula F. We prove some lemmata about this D-chain.


Lemma 1. If an atomic formula occurs as a positive (or as a negative) part in a formula Fn of the D-chain, then it also occurs in every formula Fm (m~n) of the Dchain as a positive (or as a negative) part.

Proof This follows immediately from the definition of the D-chain.

Lemma 2. If a formula Fn of the D-chain has a reducible positive (or reducible negative) part C, then the D-chain contains a reducible formula Fm (m ~n) with distinguished positive (negative) part C.

Proofby induction on the number k of ---+ and V symbols occurring in the reducible parts of Fn to the right of C. If C is the distinguished part of Fn then the assertion is satisfied with m = n. Otherwise Fn is followed by a formula Fn + I with positive (or negative) part C which contains less than k ---+ and V symbols in reducible parts to the right of C. But then the assertion follows from the induction hypothesis.

Lemma 3. If a formula Fn of the D-chain has a negative part Vx§"[x], then the D-chain contains infinitely many reducible formulas with distinguished negative part Vx§"[x].

Proof It follows from the definition of D-chain that the formula Vx§"[x] occurs as a negative part in every formula Fm (m ~n) in the D-chain. Therefore every f9rmula in the D-chain is reducible. Since the D-chain contains no axiom it follows that the D-chain is infinite. The assertion now follows from Lemma 2.

We say that a formula is a positive (or negative) part of (DC) if it occurs as a positive (or negative) part in some formula of the D-chain (DC). We now define a model.lt=<N, V) as follows:

1. VlX i : =i 2. Ifp is an n-placepredicate letter (n ~ 1) then Vp is the set ofn-tuples (iI' ... , in)

of natural numbers such thatp(lXi1 , ••• , lXi ) is a negative part of (DC). 3. If v is a negative part of (DC) then set Vv: = t. Otherwise set Vv: = f The following lemmata hold for this model.

Lemma 4. If P is an atomic formula then VP= t if, and only if, P is a negative part of (DC).

Proof Since the D-chain (DC) contains no axiom, 1. is not a negative part of (DC). For the remaining atomic formulas the assertion follows immediately from the definition of the model.

Lemma 5. IfC is a positive (or a negative) part of (DC) then VC=f(or VC= t).

Proofby induction on the length of C. 1. Suppose C is an atomic formula. Since the D-chain (DC) contains no

axiom ~[C, C] it follows from Lemma 1 that C is not both a positive and a negative part of (DC). The assertion then follows from Lemma 4.


2. Suppose C is a formula (A ---> B). If (A ---> B) is a positive part of (DC), then A is a negative and B a positive part of (DC). Then by I.H. VA = t and VB= 1 It follows that V(A ---> B)= 1 If(A ---> 1..) is a negative part of (DC), then A is a positive part of (DC). Therefore by I.H. VA =1 It follows that V(A ---> 1..) = t. If (A ---> B) is a negative part of (DC) but B is not the formula 1.., then it follows from Lemma 2 and the definition ofD-chain that (A ---> 1..) or B is a negative part of (DC). Then by I.H., VA = for VB= t. Hence V(A ---> B) = t.

3. Suppose C is a formula Vxff[x]. If Vxff[x] is a positive part of (DC), then it follows from Lemma 2 and the definition of the D-chain that some formula ff[a,] is a positive part of (DC). Then by I.H. Vff[a,] =1 Hence VVxff[x] =1 IfVxff[x] is a negative part of (DC), then it follows from Lemma 3 and the definition of the D-chain that every formula ff[a,] is a negative part of (DC). Then by I.H., Vff[a,] = t for all i EN. Hence VVxff[x] = t.

In particular from Lemma 5 we have VF=J, since Fis a positive part of (DC). This completes the proof of the principal semantic lemma.

Remark. The proof ofthe completeness theorem is non-constructive since it draws on the non-constructive notion of validity and Konig's lemma which has a nonconstructive proof. On the other hand the D-chains were constructively defined and it follows from the proof that for every deducible formula one has a standard deduction which is given, constructively, by the D-chains.

4. The Satisfiability Theorem

F or brevity we write

for

In the case n = 0 this means the formula B. A set S of formulas is said to be inconsistent if there is a finite subset {A I, ... , An}

of S such that the formula

is deducible. Otherwise the set S is said to be consistent. A set S of formulas is said to be satisfiable if there is a model At in which every

formula in the set S is true. Then we also say that S is satisfiable in At or that At is a model of the set S.

Consistency is a syntactic notion, satisfiability a semantic one. An inconsistent set of formulas is not satisfiable since if a formula

AI, ... , An ---> 1.. is deducible then by the consistency theorem it follows that for


every modelult = (!!},V): V(A 1 , ••• , An ~ ..l) = t and hence VAi=ffor at least one formula Ai from the set S.

Now, in order to prove that every consistent set of formulas is satisfiable, we proceed as for the proof of the completeness theorem. We again start with an enumeration

of all free object variables. Let S be a non-empty set of formulas. We assume that no formula in S contains

a free object variable rx i with odd index i. (In order to ensure this, one can replace each free object variable rxi which occurs in a formula of a set S 1 by rx 2i . One then obtains a set of formulas S2 of the desired sort which is satisfiable if, and only if, S 1 is satisfiable.)

The set S is at most denumerably infinite since we are only considering countably infinitely many primitive symbols. Now we assume we are given an infinite sequence of formulas

which contains all and only formulas from the set S. (This sequence must contain some formula infinitely many times if the set S of formulas is finite.)

From this sequence of formulas we define a D-chain for S as a sequence of formulas

formed as follows: DSI. The initial formula Fo of the D-chain is the formula (Ao ~ ..l). DS2. If a formula Fn of the D-chain is an axiom, then it is the last formula of

the D-chain. We then say that the D-chain has length n. DS3. If a formula Fn of the D-chain is not an axiom then the D-chain has as

the immediate successor Fn+ 1 of Fn a formula (An+ 1 ~ Gn), where Gn is determined as follows by Fo, ... , Fn.

DS3.1. If Fn is an irreducible formula, then Gn is the formula Fn. DS3.2. If Fn is a reducible formula JV[(A ~ B)] with distinguished negative

part (A ~ B), then Gn is either the formula JV[(A ~ ..l)] or the formula JV[B]. DS3.3. If Fn is a reducible formula &'[V'xff[x]] with distinguished positive

part V'xff[x] , then Gn is the formula &,[ff[rx2i+ 1]] where i is the least number such that rx 2i + 1 does not occur in Fn.

DS3.4. If Fn is a reducible formula JV[V'xff[x]] with distinguished negative part V'xff[x], then Gn is the formula (V'xff[x] ~ JV[ff[rxJ]) where i is the least number such that ff[rxJ does not occur as a negative part in any of the formulas Fo, ... , Fn if the nominal symbol * 1 occurs in ff.

Principal syntactic lemma for sets offormulas. If every D-chainfor S isjinite, then the set S is inconsistent.


Proof It follows from Konig's lemma that there are only finitely many D-chains for S. Let m be the maximal length of such a D-chain. We prove by induction on m-n:

If a formula Fn occurs in the n-th place of a D-chain for S, then the formula

is deducible. (Ifm=n, then (I) is the formula Fm .)

If Fn is the last formula of a D-chain for S, then Fn is an axiom. Then the formula (1) is also an axiom. Otherwise following Fn in the D-chains for S there are the formulas (An+ 1 --+ G~» (i= 1,2 or i= 1).

By I.H. the formula

is deducible. It follows by a basic inference (possibly using structural inferences and Theorem 4.4), that the formula (1) is also deducible. (The condition on variables for (S2)-inferences is not violated since only free object variables with even index occur in formulas of the set S.)

It now follows by taking n=O that the formula

Am' ... , Ao --+ -.L

is deducible. Hence the set S is inconsistent.

Principal semantic lemma for sets of formulas. If there is an infinite D-chain for S, then the set S of formulas is satisfiable in a model (N, V) with domain the natural numbers N.

The proof proceeds just as for the D-chain of a formula. One defines the model (N, V) as before and has the same Lemmata 1-5. Since every formula Ai of the set S occurs as a negative part in the D-chain, it follows that VAi= t for all Ai E S.

Theorem 5.4 (Satisfiability Theorem). Every consistent set offormulas is satisfiable in a model whose domain is the natural numbers N.

Proof If S is a consistent set of formulas, then by the principal syntactic lemma there is an infinite D-chain for S. The assertion now follows by the principal semantic lemma.

Theorem 5.5 (Compactness Theorem). If every finite subset of a set S offormulas is satisfiable, then S is satisfiable.

Proof If a finite set {A l' ... , An} is satisfiable, then the formula


is not valid and hence by the consistency theorem not deducible. It therefore follows from the hypothesis of our theorem, that the set S is consistent. Then by Theorem 5.4 it is satisfiable.

Remark. Theorem 5.5 establishes the compactness of a set of models in a suitable topology.

5. Syntactic and Semantic Consequences

We define the notions of syntactic consequence S ~ F and semantic consequence SF F for a (possibly empty) set S of formulas as follows:

S ~ F ("F follows syntactically from S") means: There is a finite (possibly empty) subset {Al' ... , An} of S, such that the formula

Al, ... ,An~F (n~O)

is deducible. (If S is empty, then this says that F is deducible.) SF F("Ffoliows semantically from S") means: Every model of the set S is also

a model of the formula F. (If S is empty, then this says that F is valid.) We now prove that these two notions of consequence coincide.

Theorem 5.6 (Strong Consistency Theorem). S ~ F implies SF F.

P~oof By hypothesis there is a finite subset {A 1, ... , An} of S such that the formula

is deducible, therefore by the consistency theorem it is valid. So for every model .A =(qfi, V) of Swe have VeAl> ... , An~ F)= t and VAi= t(i= 1, ... , n). Therefore VF=t. Hence we have SF F.

Theorem 5.7 (Strong Completeness Theorem). SF F impiies S ~ F.

Proof From SF F it follows that the set Su{(F ~ ..i)} of formulas is not satisfiable. By the satisfiability theorem it follows this set is inconsistent. Hence there is a finite subset {Al' ... , An} of S such that the formula

A 1 , ••• , An~ «F ~ ..i) ~ ..i) is deducible. From this formula and the deducible formula

«F~ ..1)~ ..1)~ F

by Theorem 4.6 it follows that the formula

is also deducible. Hence S ~ F.

Chapter III

Intuitionistic Predicate Calculus

Intuitionistic logic is based on constructive inferences which are not covered as in classical logic by truth values. This leads immediately to theories in which the truth of a proposition is either not determined semantically or is not in general decidable.

A calculus for intuitionistic predicate calculus was first developed by A. Heyting in order to use it to delimit the predicate logic which is adequate for the intuitionism of L. E. J. Brouwer. Corresponding semantic characterizations of this logic as were later given by E. W. Beth and S. Kripke, are essentially more complicated than the semantics of the classical predicate calculus.

In what follows we do not start from such semantics, instead we only define intuition is tic predicate calculus syntactically by means of axioms and basic inferences which are motivated in a natural way by logical inferences. As before the symbols ..l, /\, v and --> denote connectives, but they are no longer conceived as symbols for truth functions.

§6. Formalization of Intuitionistic Predicate Calculus

1. The Formal System IPt

In intuitionistic predicate calculus the connectives /\, v and the existential quantifier 3 cannot be defined in terms of ..l, --> and If as in classical predicate calculus.

We therefore use as primitive symbols: The primitive symbols of the system CP and the symbols /\, v and 3.

Consequently in addition to the defining rules for formulas of the system CP we have the following defining rules for formulas:

2.4'. If A and B are formulas, then (A /\ B) and (A v B) are also formulas. 2.5'. If ff[a] is a formula and x is a bound object variable which does not occur

in the I-place nominal form ff, then 3xff[x] is also a formula. Atomic formulas are defined as in the system CPo We use the same syntactic symbols for free and bound object variables, prop

ositional variables, formulas and atomic formulas as in the system CPo

6. Formalization of Intuitionistic Predicate Calculus 37

For brevity we write

for

We denote by r, r;, ,1, ,1i finite (possibly empty) sequences of formulas. If r =

At, ... , An (n?: 1), then r - B denotes the formula

If r is empty, then r - B denotes the formula B. r e,1 means that every formula in the sequence r is also a formula in the

sequence,1. A formula which occurs several times in r however, need only occur once in ,1. (In particular r e,1 holds for all ,1 when r is empty.)

For implication we first take as basic only the following axioms and rules of inference:

(Ax. 1) A- A. (Str) r - B f- ,1- B if re,1 holds. (Cut) r - A, A - B f- r - B.

This says that a formula r - B should only be deducible if one can get from the formulas in the sequence r to the formula B in a trivial way. We call a (Str) inference a structural inference and a (Cut) inference a cut (with cut formula A).

Restricting to formulas built up only from propositional variables by using the connective - and using (Ax. 1), (Str) and (Cut) gives the positive implicational calculus of P. Bernays.

Now for the connective 1.. we add the axiomschema

(Ax. 2) 1..- A (exfalso quodlibet)

and we fix the syntactic use of the symbols /\, v, V and :3 by two basic inferences each depending on whether these symbols are.introduced in a left or right component of a formula.

(/\ 1) A - B - C f- (A /\ B) - C ( /\ r) r - A, r - B f- r - (A /\ B) ( v 1) A - C, B - C f- (A v B) - C (vr) r-Aif-r-(AlvA2) (i=I,2) (VI) ff[a] - C f- Vxff[x] - C (Vr) r - ff[a] f- r - VXff[X]} if a does not occur (:31) ff[a] - C f- :3xff[x] - C in the conclusion (:3r) r - ff[a] f- r - :3xff[x].

38 III. Intuitionistic Predicate Calculus

We call the system with axioms (Ax. 1), (Ax. 2) and basic inferences (Str), (Cut), ( /\ 1), ... , (3r) the formal system IP 1 of intuitionistic predicate calculus.

2. The Formal System IP2

Among the basic inferences of the system IP1 the cuts

r -4 A, A -4 B ~ r -4 B

are the only inferences where a formula A may occur in the premises which need bear no formal relation to the conclusion. Thus the deducibility of a formula r -4 B may, because of a cut, depend in a roundabout way on an arbitrary formula A. However, G. Gentzen arrived at just as straightforward a pattern of inference for intuitionistic as for classical predicate calculus in which in every premise,of an inference only subformulas of the conclusion occur (where one regards ff[a] as a subformula of'v'xff[x] and 3xff[x]). One obtains a straightforward pattern of inference if one takes a suitable basic rule of inference for implication so that the cut is no longer needed as a basic inference but nevertheless still can be proved permissible.

One rule of inference for introducing implication, which corresponds to the straightforward pattern of inference, is (-41) r -4 A, B -4 C ~ r -4 (A -4 B) -4 C. Such an (-4 I)-inference is permissible in the system IPI as one sees as follows:

1) (A -4 B) -4 A -4 B is an (Ax. 1), 2) A -4 (A -4 B) -4 B follows from 1) by a structural inference, 3) r -4 A is the first premise of the (-4 I)-inference, 4) r -4 (A -4 B) -4 B follows from 3) and 2) by a cut, 5) B -4 C is the second premise of the (-4 I)-inference, 6) r -4 (A -4 B) -4 C follows from 4) and 5) by a cut. If one adds (-4 1) as a basic rule of inference then (Ax. 1) can be restricted to

(Ax. 1A) P -4 P

where P is an atomic formula, since then every formula A -4 A is deducible (without the use of cuts). Thus, for example, from A -4 A and B-4 B by an (-41)inference we get A -4 (A -4 B) -4 B and by a structural inference (A -4 B-4 (A -4 B). (Ax. 2) may also be restricted to

(Ax.2A) ..l-4 P.

We denote by IP2 the formal system with axioms (Ax. 1A), (Ax. 2A) and basic inferences (Str), (-41), ( /\ 1), ... , (3r). This system has the desired property that only subformulas of the conclusion occur in each premise of a basic inference. This has certain advantages for syntactic investigations. For example it is immediately apparent that the system IP2 is consistent since no atomic formula has the form of an axiom or a conclusion of a basic inference of the system IP2 it follows that no


atomic formula is deducible in IP2. In order to prove this is also true for IPI one has to show that an atomic formula cannot be shown to be deducible using a cut. This is not immediately obvious, since the conclusion of a cut can be an atomic formula.

In IP2 it turns out that the formulas A - A and 1.._ A are deducible and the cut is a permissible inference whence it follows that precisely the same formulas are deducible as in IPI. We shall not carry out the proof of the deducibility of A - A and 1.._ A and of the permissibility of cut in IP2 since we shall pass to another formal system IP3 and give the corresponding proof for it in §7.

3. Left and Right Parts of Formulas

The notions of "positive" and "negative part" of a formula which we used in classical predicate calculus refer to valuations of formulas and truth values and are therefore not available for intuitionistic predicate calculus. Instead of using these for the basic inferences we turn to certain left and right parts offormulas which are defined as follows. Every formula can be uniquely written in the form

r-B

where B is not of the form (B1 - B2). The left parts of such a formula are all f.ormulas ofthe sequence rand therightpartsaretheformulaBand,ifr=A 1, ... , An (n ~ 1), so too are An- B and all formulas Ai - ... - An- B (i= 1, ... , n-I). Logical symbols are introduced into a left part of the conclusion by (- 1), ( "1), ( v 1), (VI) and (31) basic inferences, and into a right part of the conclusion by ( "r), ( v r), (Vr) and (3r) basic inferences.

In order to be able to do without the ( "1) inference as a basic inference we make the notion of "left part" somewhat wider in that we also regard A and Bas left parts of a formula F if (A "B) is a left part of F.

In order to characterize left and right parts of a formula we use L-forms (left partforms) and R-forms (right partforms) which are inductively defined as I-place nominal forms as follows:

LRF 1. *1 is an R-form. LRF 2. If fYl is an R-form and B is a formula, then fYl[(*l - B)] is an L-form

and fYl[(B - * 1)] is an R-form. LRF 3. If 2is an L-form and Ba formula then 2[(*1 "B)] and 2[(B "*1)]

are L-forms. We have the following consequences of this inductive definition: LRF 4. No R-form is an L-form. LRF 5. If fYl is an R-form and tilt is an L-form (or an R-form) then ~[tiIt] is an

L-form (or an R-form). LRF 6. If tilt is an L-form or an R-form and A is a formula then tiIt[A] is also

a formula. We now define : A formula A is said to be a left part (or a right part) ofa formula

Fifthere is an L-form (or an R-form) tilt such that tiIt[A] is the formula F. From this


definition by means of the defining rules LRF I-LRF 3 we obtain the following inductive characterization of left and right parts of a formula F.

LR I. F is a right part of F. LR 2. If(A --. B) is a right part of F, then A is a left part and B a right part of F. LR 3. If (A A B) is a left part of F, then A and B are also left parts of F. As corollaries we have: LR 4. If F is not an implication, that is not a formula of the form (A --. B),

then F is the unique right part of F. LR 5. If F is an implication (A --. B), then the right parts of F are the formula

F and the right parts of B. LR 6. Every formula Fhas precisely one right part which is not an implication.

We call this the end part of the formula F. LR 7. The end part of a formula F is also the end part of every right part of F. By a D-form (double form) we mean a 2-place nominal form [f) such that

[f)[*1' B] is an L-form and [f)[A, *1] is an R-form. For syntactic symbols we use

for L-forms, for R-forms, for D-forms.

A left part of a formula is said to be minimal if it is not of the form (A A B). We generalize the structural inference (Str) thus: F ~ G ("G follows structurally

from F") means that F and G have the same end part and every minimal left part of Falso occurs as a left part ofG. (Obviously r --. B ~ LI--. Bholdsif r --. B I- LI --. B is an (Str)-inference.)

4. The Formal System IP3

We obtain a formal system for intuitionistic predicate calculus in which neither cuts nor structural inferences are needed if the axiom schemata and basic rules of inference of the system IP 2 are modified in such a way that they are invariant under structural inferences in the following sense: Every formula which follows from an axiom by a structural inference is likewise to be an axiom and every formula which follows from the conclusion of a basic inference by a structural inference is to be the conclusion of a corresponding basic inference whose premises are to be derivable from the premises of the first basic inference.

We get such an invariance under (Str)-inferences if, for example we replace the axiom schema (Ax. IA) by

and the basic rule of inference (--. I) by


In order to get this invariance with respect to F ~ G too we replace (Ax. lA) by ~[P, P] and (----+ I) by

~[(A ----+ B), A], ~[B, C] f- ~[(A ----+ B), C]

We correspondingly modify the remaining axioms and basic inferences of the system IP2.

The formal system IP3 of the type we want has the same formal language as the systems IP 1 and IP2.

The axioms of the system IP3 are all formulas of the following forms

(Ax. I) ~[P, P] (Ax. II) 2[1-].

The basic inferences of the system IP3 are all inferences of the following sort:

(----+ L) ~[(A ----+ B), A], ~[B, C] f- ~[(A ----+ B), C] (/\ R) 9l[A], 9l[B] f- 9l[(A /\ B)J (v L) 2[A], 2[B] f- 2[(A v B)J (vR) 9l[A;]f-9l[(AIVA2)] (i=1,2) (\t'L) 2[('1xff[x] /\ ff[a])] f- 2['1xff[x]] ('1R) 9l[ff.[a]] f- 9l['1Xff[X]]} with a condition on (3L) 2[ff[a]] f- 2[3xff[x]] variables (3R) 9l[ff[a]] f- 9l[3xff[x]].

The free object variable denoted by a in the premises of the ('1R) and (3L) inferences is said to be the eigenvariable of the given inference. The condition on variables for these inferences is: The eigenvariable must not occur in the conclusion.

The left part denoted by (A ----+ B) in the conclusion of an (----+ L)-inference is called the prinCipal part, the right part of the conclusion denoted by C is called the secondary part of the given basic inference. For any other basic inference the principal part is the left or right part of its conclusion shown. In every case the principal part ofa basic inference is either a minimal left part or the end part of the conclusion. For ----+ one only has a basic L-inference with principal left part, for /\ only a basic R-inference with principal right part and for v, '1, 3 one basic Linference and one basic R-inference.

By the degree of a formula we mean the number of ----+, /\, v, V' and 3 symbols occurring in the formula.

It follows easily from the results of the next section that precisely the same formulas are deducible in the system IP3 as in the system IPl. However, the system IP3 has the advantage that induction on deductions is particularly simple and transparent since here the derivation is direct and also no structural inference occurs as a basic inference.


§7. Deducible Formulas and Permissible Inferences in the System IP3

1. Generalizations of the Axioms

Theorem 7.1. Every formula ~[C, C] is deducible.

Proofby induction on the degree of the formula C. 1. Suppose C is an atomic formula. Then ~[C, C] is an axiom. 2. Suppose C is a formula (A ---4 B). Since ~[(A ---4 B), (*1 ---4 *2)] and

~[*1' (A ---4 *2)] are D-forms, by the LH. (induction hypothesis) I- ~[(A ---4 B), (A ---4 A)] and I-~[B, (A ---4 B)]. Using an (---4 L)-inference with the D-form ~[*1' (A ---4 *2)] we get I-~[(A ---4 B), (A ---4 B)].

3. Suppose C is a formula (A /\ B). Since ~[(*1 /\B), *2] and ~[(A /\ *1)' *2] are D-forms, by LH. we have I-~[(A /\ B), A] and I-~[(A /\ B), B]. ,Since ~[(A /\ B), * 1] is an R-form, by an ( /\ R)-inference we obtain I-~[(A /\ B), (A /\ B)].

4. Suppose C is a formula (A v B). By LH. we have I-~[A, A] and I-~[B, B]. Using ( v R)-inferences we get I-~[A, (A v B)] and I-~[B, (A v B)]. Then by an ( v L)-inference we have I-~[(A v B), (A v B)].

5. Suppose C is a formula V'xff[x] or 3xff[x]. We choose a free object variableawhichdoesnotoccurin~[C, C]. ByLH. we have I-~[(V'xff[x] /\ ff[a]), ff[a]] and I-~[ff[a], ff[a]]. Using an (V'L)-inference and an (3R)-inference we have I-~[V'xff[x], ff[a]] and I-~[ff[a], 3xff[x]]. Using an (V'R)-inference and an (3L)-inference we obtain I- ~[V'xff[x], V'xff[x]] and I-~[3xff[x], 3xff[x]],

2. Weak Inferences

Theorem 7.2 (Substitution rule). If a1 does not occur in the nominalform ff then


The proof proceeds by induction on the deduction as for Theorem 4.2.

Theorem 7.3. The following are weak inferences for the elimination of logical symbols

(---4 E) 2'[(A 1 ---4 A 2)] I- 2'[A2] (/\ E) 9I![(A1/\ A 2)] I- 9I![AJ (i= 1, 2) (vE) 2'[(A1VA2)]I-2'[AJ (i=1,2) ('IE) 9I![V'xff[x]] I- 9I![ff[a]] (3E) 2'[3xff[x]] I- 2[ff[a]].

1. Proof for (---4 E), (/\ E) and (v E). Suppose OU[(A1 x A 2)] is a formula 2[(A1 ---4 A 2)], 9I![(A1/\ A 2)] or 2[(AJ v A 2)] which is deducible with order n.

7. Deducible Formulas and Permissible Inferences in the System IP3 43

We prove by induction on the deduction that Olt[AJ is then deducible with order ~n. For this let i=2 if Olt[(Al x A z)] is the formula 'p[(Al ~ A z)], otherwise i= lor i=2.

1.1. Suppose Olt[(Al x A z)] is an axiom. Then Olt[AJ is also an axiom. 1.2. Suppose Olt[(Al x A z)] was obtained by a basic inference whose principal

part is the left or right part (Al x A z) shown. Then Olt[AJ is a premise of this basic inference and therefore is deducible with order < n.

1.3. Suppose Olt[(Al x A z)] was obtained by an (~L)-inference whose secondary part contains the part (Al x A z) shown. The premises of this (~ L)inference are of the form E&[(Bl ~ B z), B l ] and E&[Bz, OltO[(Al x A z)]] where Olt[AJ is the formula E&[(Bl ~ B z), Olto[AJ]. Since E&[Bz, OltO[(Al x A z)]] is deducible with order <n, by I.H. E&[Bz, Olto[AJ] is also deducible with order <no Since E&[(Bl ~ B z), B l ] is also deducible with order <n it follows by an (~ L)inference that E&[(Bl ~ B z), Olto[AJ] is deducible with order ~n.

1.4. Suppose OU[(Al x A z)] was obtained in some other way by a basic inference. Then every premise of this basic inference is of such a form Oltj[(Al x A z)] that

Oltj[AJ U=1,2orj=1) f- Olt[AJ

is also a basic inference. By I.H. Oltj[AJ is deducible with order <no Using the given basic inference it follows that Olt[AJ is deducible with order ~ n.

2. Prooffor (V'E) and (3E). Suppose Olt[Qx$'[x]] is a formula 9l[V'x$'(x]] or 'p[3x$'[x]], which is deducible with order n. We prove by induction on the deduction that Olt[$'[a]] is then deducible with order ~ n.

2.1. Suppose Olt[Qx$'[x]] is an axiom. Then Olt[$'[a]] is also an axiom. 2.2. Suppose Olt[Qx$'[x]] was obtained by a basic inference whose principal

part is the left or right part Qx$'[x] shown. The premise Olt[$'[a l ]] of this basic inference is deducible with order <no Since the eigenvariable a l of the basic inference does not occur in the nominal form Olt[$'] it follows by Theorem 7.2 that Olt[$'[a]] is also deducible with order <no

2.3. Suppose Olt[Qx$'[x]] was obtained in some other way by a basic inference. We choose a free object variable a l which does not occur in the nominal form Olt[$'] and is not the eigenvariable of the basic inference. By I.H. it follows as in 1.3 and 1.4 that Olt[$'[a l ]] is deducible with order ~ n. Then by Theorem 7.2 Olt[$'[a]] is also deducible with order ~ n.

Theorem 7.4 (Structural rule). If F~ G then Ff- G is a weak inference.

Proof by induction on the deduction. Suppose F is deducible with order nand F~ G holds.

1. Suppose Fis an axiom. Then from F~ G it follows that G is also an axiom. 2. Suppose Fis a formula E&l[(A ~ B), C] which was obtained by an (~ L)

inference from E& l[(A ~ B), A] andE& l[B, C]. Then GisaformulaE&z[(A ~ B), E] where Eis the end part of F. Let E&o[B, E] be the formula obtained fromE& l[B, C]


by replacing every leftpart (A ~ B) of this formula by B. By Theorem 7.3 !?JJo[B, E] is deducible with order <no From FflG we have !?JJl[(A~B),A]fl !?JJ2[(A ~ B), A] and !?JJo[B, E] fl !?JJ 2[B, E]. Then by I.H. !?JJ 2[(A ~ B, A] and !?JJ 2[B, E] are deducible with orders <no Using an (~ L)-inference it follows that !?JJ2[(A ~ B), E] is deducible with order ~n.

3. Suppose F is a formula 9f!l[(A l /\ A2)] which was obtained by an (/\ R)inference from 9l l [A l ] and 9f!1[A2]. Then G is a formula 9l2[(A l /\A 2)] where 9f!l[AJ fl9f!2[A;] (i= 1,2). By I.H. 9f!2[AJ is deducible with order <no Using an (/\ R)-inference it follows that 9l2 [(A l /\A 2 )] is deducible with order ~n.

4. Suppose F is a formula 2\ [(Al v A2)] which was obtained by an (v L)inference from 2'l[A l ] and 2'1[A2]. Then G is a formula 2'2 [(A 1 v A 2)]. For i= 1,2 let Fi be the formula obtained by replacing every left part (Al v A2) in 2'l[AJ by Ai' By Theorem 7.3 Fi is deducible with order <no From Ffl Gwehave Fi fl2'2[AJ. Therefore by I.H. 2'2[AJ is deducible with order <no Using an (v L)-inference it follows that 2'2 [(A 1 v A 2 )] is deducible with order ~iI. ,

5. Suppose F was obtained by an ( v R)-inference, (VL)-inference or (3R)inference. Then the assertion follows as in case 3.

6. Suppose F was obtained by a (VR)-inference or (3L)-inference with eigenvariable a. We choose a free object variable a l which occurs in neither Fnor G. Let Fl be the formula obtained by replacing the eigenvariable a in the premise of the basic inference by a l . By Theorem 7.2 Fl is deducible with order <no Fl f- Fis a basic inference with eigenvariable a l . Now the assertion follows as in case 3 or 4.

3. More Permissible Inferences

Theorem 7.5. 2'[g;[a]] f- 2'[Vxg;[x]] (provided x does not occur in g;) is a permissible inference.

Proof We have 2'[g;[a]] fl2'[(Vxg;[x] /\ g;[a])]. Therefore by Theorem 7.4 from f- 2'[g;[a]] we have f- 2'[(Vxg;[x] /\ g;[a])]. Using a (VL)-inference we get f- 2'[Vxg;[x]].

Theorem 7.6 (Cut rule). 9f![A], (A ~ B) f- 9f![B] is a permissible inference.

Proofby induction on the degree of the formula A. 1. Suppose A is a formula (Al /\A 2). By Theorem 7.3 from hgj?[(Al /\A 2)]

we have f-9l[A l ] and f-9f![A2l By Theorem 7.4 from f-«Al /\A2)~ B) we have f-(A2 ~ (A2 ~ B)). By the I.H. using f-9l[A l ] we have f-9f![(A2 ~ B)]. By Theorem 7.4 we have f- A2 ~ 9l[B]. By I.H. using f-9f![A 2] we get f-9l[9f![B]]. By Theorem 7.4 there follows f-9f![B].

2. Suppose A is an atomic formula or a formula (Al v A2) or 3xg;[x]. In this case we perform a subsidiary induction on the order of a deduction of 9f![A].

2.1. Suppose 9f![A] is an axiom. 2.1.1. If 9f![A] is an (Ax. I) then A is an atomic formula and 9l[B] is a formula

!?JJ[A, B]. By Theorem 7.4 from f-(A ~ B) we have f-!?JJ[A, B]. 2.1.2. If 9f![A] is an (Ax. II) then 9l[B] is also an (Ax. II).


2.2. Suppose .?l[A] was inferred by a basic R-inference. Then A is either (AI v Az) or 3x~[x].

2.2.1. If A is the formula (AI v A2) then the basic R-inference has a premise 9i![A;J (i = 1 or i = 2). By Theorem 7.3 from f-«A 1 v A2) ---> B) we have f-(Ai ---> B). By I.H. using 9i![A;J we have f-.?l[B].

2.2.2. If A is the formula 3x~[x] then the basic R-inference has a premise 9i![~[a]]. By Theorem 7.3 from f-(3x~[x] ---> B) we have f-(~[a] ---> B). By I.H. using f-9i![F[aJJwe have f-9i![B].

2.3. Suppose .?l[A] was inferred by an (---> L)-inference. This inference has as premises .@[(CI ---> C2), C1 ] and .@[Cz, 9i!o[A]] where 9i! is the R-form .@[(CI ---> C2), 9i!o]. From f-.@[C2,.?lo[A]] and f-(A ---> B) by the S.I.H. (subsidiary induction hypothesis) we obtain f-.@[C2 , 9i!o[B]]. By an (---> L)-inference using f-.@[(CI ---> C2), CI] we have f-.@[(CI ---> C2), .?lo[B]], i.e. f-.?l[B].

2.4. Suppose 9i![A] was inferred by an (v L)-inference, (VL)-inference or (3L)-inference. In the case of an (3L)-inference we may assume by Theorem 7.2 that the eigenvariable of this inference does not occur in B. Then each premise of the basic L-inference is of the form .?l;[A] so

9i!;[B] (i= 1, 2 or i= 1) f-9i![B]

is also a basic L-inference. From f-.?li[A] and HA ---> B) by the S.I.H. we therefore have f-9i!;[B]. Using the given basic L-inference f-9i![B] follows.

3. Suppose A is a formula (AI ---> A2) or Vx~[x]. For this case we use a subsidiary induction on the order of a deduction of (A ---> B).

3.1. Suppose (A ---> B) is an axiom. Since A is a minimal left part of (A ---> B) which is not an atomic formula then 9i![B] is also an axiom.

3.2. Suppose (A ---> B) was inferred by a basic inference with principal term A. 3.2.1. Suppose A is the formula (A I ---> A 2). Then the basic inference has

premises «AI ---> A2)---> 9i!O[AI]) and (A2 ---> 9i!o[C]) where B is the formula 9i!o[C]. By the S.I.H. from f-9i![(AI ---> A2)] and f-«AI ---> A2)---> .?lo[Ad) we have hJ4![9i!o[AI]]. By I.H. from f-9i![(AI ---> A2)] and f-(A2 ---> 9i!o[C]) we have f-9i![(AI ---> 9i!o[C])]. By Theorem 7.4 we have f-(AI ---> .?l[9i!o[C]]). Using f-9i![.?lO[AIJJ there follows by I.H. f-.?l[.?lo[9i![.?lo[C]]]]. By Theorem 7.4 we have f-9i![9i!o[C]], that is, f-9i![B].

3.2.2. Suppose A is the formula Vx~[x]. Then the basic inference has a premise «Vx~[x] /I. ~[a]) ---> B). By Theorem 7.4 the formula (Vx~[x]---> (~[a] ---> B» is also deducible with an order smaller than that of the formula (Vx~[x] ---> B). Using f-.?l[Vx~[x]] therefore, f-.?l[(~[a] ---> B)] follows by the S.I.H. By Theorem 7.4 there follows f-(~[a] ---> 9i![B]). From f-9i![Vx~[x]] by Theorem 7.3 we have f-9i![~[a]]. Using f-(~[a] ---> .?l[B]), by I.H. we have f-.?l[9i![B]]. By Theorem 7.4 we have f-9i![B].

3.3 Suppose (A ---> B) was inferred by a basic inference whose principal part is contained in B. Then by Theorem 7.2 we may assume that an eigenvariable of the basic inference is not a free object variable occurring in 9i!. Then each premise of the basic inference is of the form (A ---> B i ) so

9i![B;J (i= 1, 2 or i= 1) f-9i![B]


is a basic inference. From 1-8l[A] and I-(A -> B;) by the S.I.H. we have I-Bl[BJ. Using the given basic inference we have I-Bl[B].

Corollary to Theorem 7.6. Theformal system IP3 is s-consistent, that is, there is no formula A such that A and (A -> l..) are both deducible in IP3.

Proof By Theorem 7.6

A, (A -> l..) I- l..

is a permissible inference in the system IP3. Were A and (A -> l..) deducible then l.. would also be deducible. But this is not the case since l.. is not an axiom nor does it occur as the conclusion of a basic inference.

Remark. As will be shown in §8, every formula deducible in IP3 is also deduc;ible in CPo Hence the s-consistency of the system IP3 also follows from the s-consistency of the system CPo

4. Special Features of Intuitionistic Logic

Theorem 7.7. 1. A formula (A v B) is deducible only if A is deducible or B is deducible.

2. A formula 3xff[x] is deducible only if there is a free object variable a such that ff[a] is deducible.

Proof 1. (A v B) is not an axiom. The only basic inferences with (A v B) as conclusion are the ( v R)-inferences with premise A or B. Therefore I-(A v B) holds only if I- A or I- B holds.

2. 3xff[x] is not an axiom. The only basic inferences with 3xff[x] as conclusion are the (3R)-inferences with a premise ff[a]. Therefore 1-3xff[x] holds only if there is an a with I-ff[a].

Corollaries to Theorem 7.7. The formulas (v v (v -> l..» and ('Ix p(x) v 3x(P(x) -> l..», where p is a one-place predicate letter, which are valid in classical predicate calculus are not deducible in IP3.

Proof 1. Were (v v(v-> l..» deducible then by Theorem 7.7 v or (v-> l..) would be deducible. But this is not the case since neither v nor (v -> l..) occurs as an axiom or as the conclusion of a basic inference.

2. Were ('Ix p(x) v (3x(p(x) -> l..» deducible th~n by Theorem 7.7 'Ix p(x) or 3x(p(x) -> l..) would be deducible. By Theorems 7.3 and 7.7 pea) or (p(a) -> l..) would then be deducible for some free object variable a. But this is not the case since neither pea) nor (p(a) -> l..) occurs as an axiom or as the conclusion of a basic inference.


5. Properties' of Negation

As in the system CP we define ,A : =(A -> .1.).

Theorem 7.S. f»[A, B] f- f»[,B, ,A] is a permissible inference.

Proof By Theorem 7.4 from f»[A, B] we have f-f»[(B-> .1.), (A -> B)l By an (-> L)-inference using the axiom f»[.1., (A -> .1.)] we have 1-f»[(B -> .1.), (A -> .1.)], hence 1-f»[,B, ,Al

Theorem 7.9. The following formulas are deducible

(1) (A-> "A) (2) (".1. -> .1.) (3) (",A -> ,A) (4) «,A -> A)-> , ,A)

Proof 1. By Theorem 7.1 we have I-«A -> .1.) -> (A -> .1.». By Theorem 7.4 there follows I-(A -> «A -> .1.) -> .1.), that is, I-(A -> "A).

2. From the axioms «(.1.-> .1.)-> .1.)-> (.1.-> .1.» and (.1.->.1.) by an (-> L)inference we have f-«(.1. -> .1.) -> .1.) -> .1.), that is, 1-(".1. -> .1.).

3. By (1) we have f-(A -> , ,A). By Theorem 7.8 there follows f-{",A -> ,A).

4. By Theorem 7.1 we have f-«,A -> A) -> (,A -> ,A». From (1) we have f-(A->('A->.1.». By an (->L)-inference there follows f-«,A->A)-> (,A -> .1.), that is, f-«,A -> A) -> , ,A).

6. Syntactic Equivalence

Let ~F denote that the formula F is deducible in the system IP3. Let ALB denote that (A -> B) and (B-> A) are deducible in IP3. Let A k B denote that A ~ Band B ~ A hold in IP3.

Theorem 7.10. L is an equivalence relation which is compatible with ~, k and->, that is, the following hold:

(1) ALA. (2) ALB implies BLA. (3) ALB and BL C imply ALe. (4) ALB and ~A imply ~B. (5) A kB implies ALB. (6) ALB implies (A -> C) L (B-> C) and (C -> A) L (C -> B).

Proof (1) holds by Theorem 7.1. (2) holds by the definition of L. (3) and (4) follow from Theorem 7.6.


. From I-(A -+ A) and A ~ B by Theorem 7.4 we have I-(A -+ B). This gives (5). From I-(A -+ B) we have I-(A -+ (C -+ B». Using 1-« C -+ A) -+ (C -+ C» we

obtain I-«C-+ A)-+ (C-+ B» by an (-+ L)-inference. From I-(B-+ A) there follows I-«A-+ C)-+ (B-+ A». Using I-(C-+ (B-+ C» we obtain I-«A-+ C)-+ (B-+ C». This gives (6).

§8. Relations between Classical and Intuitionistic Predicate Calculus

1. Embedding IP3 in CP

We now regard every formula of the system IP3 as also being a formula of the system CP where A, v and 3 are defined as in §4.4. We need the following lerpmata in order to show that every formula deducible in IP3 is also deducible in CPo

Lemma 1. Every R-form is a P-form, every L-form is an N-form.

Proofby induction on the length of the R-form or L-form. 1. The R-form * 1 is a P-form. 2. IfBl is a P-form then: Bl[(*1 -+ B)] is an N-form, Bl[(B-+ *1)] is a P-form. 3. If 2 is an N-form then: 2'[(*1 -+ 1..)] is a P-form, 2[((*1-+ (B-+ 1..»-+

1..)] is an N-form, 2[«B-+ *d-+ 1..)] is a P-form and 2'[((B-+ (*1-+ 1..»-+ 1..)] is an N-form. By the definition of A, 2[(*1 A B)] and 2'[(B A *1)] are therefore N-forms.

Lemma 2. Every D-form is an NP-form.

Proof This follows from Lemma 1.

Lemma 3. Thefollowing are permissible inferences in the system CP:

(I) ,q[(A -+ B), A], ,q[B, C) I- ,q[(A -+ B), C], (2) %[Vxff[x] A ff[a])] I- %[Vxff[x]].

Proof 1. By Theorem 4.3, from I- ,q[(A -+ B), A] by an inversion inference we have I- ,q[(A -+ 1..), A]. By Theorem 4.4, using a structural inference we obtain I- ,q[(A -+ 1..), C). From this and I- ,q[B, C) an (SI)-inference gives I- ,q[(A -+ B), C).

2. Since (Vxff[x] A ff[a]) is defined in PC to be the formula «Vxff[x]-+ (ff[a] -+ 1..» -+ 1..), from I-%[(Vxff[x] A ff[a])] by Theorem 4.4 a structural inference we have 1-«1..-+ 1..)-+ (Vxff[x]-+ %[ff[a]]». Since (1..-+ 1..) is deducible by Theorem 4.6 we have I-(Vxff[x] -+ %[ff[a]]). By an (S3)-inference we have I-%[Vxff[x]].

Theorem 8.1. Every formula deducible in IP3 is deducible in CPo

8. Relations between Classical and Intuitionistic Predicate Calculus 49

Proof By Lemmata 1 and 2 every axiom of the system IP3 is an axiom of the system CPo By Lemmata 2 and 3 every (-4 L)-inference is permissible in CPo By Lemma 1 every ('v'R)-inference is an (S2)-inference. By Lemmata 1 and 3 every ('v'L)-inference is permissible in CPo By Lemma 1 and Theorem 4.8 all the remaining basic inferences of the system IP3 are permissible in CPo

2. Interpretation of CP in IP3

Inductive definition of a formula F for each formula F of the system CPo

1. ..L:=..L 2. P: = , ,p for every other atomic formula P 3. (A -4 B) : = (A -4 B)

4. If $'[a] = $'[a] , where a does not occur in the nominal forms $' and $'

then 'v'x$'[x] : = 'v'x$'[x].

Lemma 4. (,' A -4 A) is deducible in IP3.

Proofby induction on the degree of the formula A.

1. Suppose A is an atomic formula. Then we have ~(, ,A -4 A) by Theorem 7.9 (2) and (3) .

• 2. SupposeAisaformula(A l -4 A2). ByTheorem7.1 we have fL«A l -4 A2)-4 (Ai -4 A2». By two applications of Theorem 7.8 we have ~(, ,(Ai -4 A2)-4 (Ai -4 "A2»' By I.H. we have ~(, ,A2 -4 A2)' By Theorem 7.6 there follows ~(, ,(Ai -4 A2) -4 (Ai -4 A2» and therefore ~(, ,A -4 A).

3. Suppose A is a formula 'v'x$'[x]. Suppose $'[a] = $'[a] , where a occurs in neither $' nor $'. By Theorem 7.1 we have ~($'[a] -4 $'[ a]). Therefore by Theorem 7.5 we have ~('v'x$'[x] -4 $'[a]). Two applications of Theorem 7.8 yield ~(, ,'v'x$'[x] -4 ,,$'[a]). By I.H. we have ~(, ,$'[a] -4 $'[a]). By Theorem 7.6 therefore ~(, ,'v'x$'[x]-4 $'[a]). From an ('v'R)-inference we have ~(, ,'v'x$'[x] -4 'v'x$'[x]) and thereforej-!(, ,A -4 A).

Proof By Theorem 7.9 (1) and Lemma 4 we have BL, ,B. Therefore (,A-4BL(,A-4" B). Since (,A-4" B)d:(, B-4 "A) holds we have (,A-4B)L(,B-4 "A). Using "ALA we obtain (,A-4B)L (,B-4A).

Lemma 6. (,A -4 A) LA:.

Proof By Theorem 7.9 (4) we have ~« ,A -4 A) -4 "A), and by Lemma 4 fL(, ,A -4 A). Therefore by Theorem 7.6 we have fL«, A -4 A) -4 A). By Theorem 7.1 we also havefL(A-4 (,A-4 A».


Lemma 7. For every P-form flJ there is an R-form f!Il such that gt'[AJ"!' f!Il[AJ holds for every formula A.

Proofby induction on the length of gt'. I. Suppose flJ is the P-form * l' Then the assertion holds for f!Il: = * l'

2. Suppose flJ is a P-form gt'o[(B- *d]. Then by I.H. there is an R-form fJto such that ~~f!Ilo[CJ for every formula C. Now gt'[AJ is gt'o[(B- A)J hence gt'[AJ ~ f!Ilo[(B - A)J, that is gt'[AJ ~ f!IloWl- A)]. The assertion is therefore satisfied by f!Il: = f!IloWl- * d]'

3. Suppose flJ is a P-form .AT(*1 - .l)]. Then JV is an N-form flJO[(*1 - B)J and therefore flJ is the P-form gt'0[«*1 - .l) - B)]. By I.H. there is an'R-form f!Ilo such that gt'0[C] ~ f!Ilo[CJ for every formula C. Now gt'[AJ is gt'o[((A - .l) - B)J hence gt'[AJ L f!Ilo[((A - .l) - B)J, that is gt'[AJ L f!Ilo[((A - .l) - B)]. Since f!Ilo[((A - .l) - B)] k( IA - f!Ilo[B]) it follows that gt'[AJ"!' (,A - f!Ilo[BJ). By Lemma 5 we have flJ[AJ"!' (,f!IlO[B] - A). The assertion is therefore satisfied by f!Il:=(,f!IlO[B]- *d·

Lemma 8. For every N-form JV there is an L-form f£' such that JV[AJ ,.!,f£'[AJfor every formula A.

Proof JV is an N-form gt'[( *1 - B)]. By Lemma 7 there is an R-form f!Il such that JV[AJ"!' f!Il[(A - B)]. The assertion is therefore satisfied by f£': = f!Il[( * 1 - B)].

Theorem 8.2 (Interpretation Theorem). If F is deducible in CP then F is deducible in IP3.

Proofby induction on the deduction. 1. Suppose Fis an axiom .2[P, P]. By Lemmata 7 and 8 there is an R-form f!Il

and an L-form f£' such that F,.!, f!Il[PJ and F L f£'[P]. Hence (P - f!Il[PJ) L (p - f£'[PJ). By Theorem 7.1 we have f-!(P - f!Il[PJ). Hence f-!(P - f£'[PJ). From Theorem 7.4 we have f-!f£'[P]. Since F L f£'[PJ, we have f-!F.

2. Suppose F is an axiom JV[.l]. By Lemma 8 there is an L-form f£' such that F L f£'[.l]. Since f£'[.l J is an axiom of the system IP3, we have f-!F.

3. Suppose F is a formula JV[(A - B)J obtained by an (SI)-inference from JV[A -.lJ and JV[B]. By Lemma 8 there is an L-form f£' such that F ~ f£'[(A _ B)], JV[(A - .l)J"!' f£'[(A - .l)J and JV[BJ L f£'[B]. By I.H. therefore f-!f£'[(A - .l)J and f-!f£'[B]. By Theorem 7.4 we have f-!«A - B)(IA- f£'[(A- B)J» and f-!(B- (IF- f£'[(A- .8)J». UsingF L f£'[(A-.8)J we have f-!«A - .8) - (I A - F» and f-!(B - (IF - F». By Lemma 5 we have f-!«A-.8)- (IF- A». Using an (-L)-inference we obtain ~«A- B)(IF - F». By Lemma 6 we have f-!«A - B)- F). Using F Lf£'[(A - B)J we obtain f-!«A - .8) - f£'[(A -.8)]. By Theorem 7.4 we have f-!f£'[(A - .8)J, hence f-!F.

4. Suppose F is a formula gt'[Yxff[xJJ obtained by an (S2)-inference from gt'[ff[aJ]. By Lemma 7 there is an R-form f!Il such that F Lf!Il[Yxff[xJJ and

9. The Interpolation Theorem 51

&'[~[a]],.l ~[~[a]]. By I.H. we have fL~[~[a]]. Now a does notoccurin~[~]. By an (VR)-inference we have fl~[Vx~[x]], hence flF.

5. Suppose F is a formula %[Vx~[x]] obtained by an (S3)-inference from (Vx~[x] -4 %[~[a]]). By Lemma 8 there is an L-form fI' such that F ,.lfl'[Vx~[x]] and (Vx~[x] -4 %[~[a]]),.l(Vx~[x] -4 fI'[~[a]]). By I.H. we have fl(Vx~[x] -4 fI'[~[a]]). By Theorem 7.4 we have flfl'[(Vx~[x] A ~[a])]. By an (VL)-inference we have flfl'[Vx~[x]], hence flF.

Let ~F denote that the formula F is deducible in the system CP. Let A~B denote that (A -4 B) and (B-4 A) are deducible in CPo

Theorem 8.3. A ~ A.

Proofby induction on the degree of the formula A. «(*1-4.1)-4.1)-4*2) and (*1-4«*2-4.1)-4.1» are NP-forms. Con

sequently ("-, --, P -4 P) and (P -4 --, --, P) are axioms of the system CP. Hence p £ P for every atomic formula P. For the remaining formulas the assertion follows from the induction hypothesis since two formulas are classically equivalent if they only differ from each other by classically equivalent components.

§9. The Interpolation Theorem

As a sort of converse of the cut rule we have an interpolation theorem which was first formulated and proved for classical predicate calculus by W. Craig. Here we prove the corresponding theorem for the formal system IP3 and then show that this yields the interpolation theorem for the system CP.

1. Interpolation Theorem for the System IP3

If X is a formula or a nominal form then we write <X) for the set of free object variables, sentential variables and predicate letters occurring in X.

Interpolation Theorem. For every formula ~[B] deducible in IP3 there is aformula A with the following properties:

1. ~[A] and (A -4 B) are deducible in IP3. 2. A contains only free object variables, sentential variables and predicate letters

which occur in both ~ and B, that is to say, <A)£ <qt)n<B).

We call such a formula an interpolant between qt and B.

Proof of the interpolation theorem. Suppose the formula ~[B] was deduced with order n. We prove, by induction on the order n of the deduction, that there is an interpolant between ~ and B.


1. Suppose ~[B] is an axiom. 1.1. Suppose B is an axiom. Then ~[(1.~ 1.)] and «1.~ 1.)~ B) are

axioms. Then (1. ~ 1.) is an interpolant between ~ and B. 1.2. Suppose ~[B] is an (Ax. I) and B is not an axiom. Then ~ is an R-form

.@[P, *1] where P is the end part of B. In this case ~[P] and (P~ B) are axioms. P occurs in ~ as well as in B. Hence P is an interpolant between !Yt and B.

1.3. Suppose ~[B] is an (Ax. II) and B is not an axiom. Then 1. is an interpolant between !Yt and B.

2. Suppose ~[B] was obtained by a basic inference whose principal part is contained in B. Each premise of this basic inference is of the form ~[Bi] (i = 1, 2 or i= 1). ByI.H. there exist Ai such that I- ~[Aa, I-(Ai ~ Bi)and <Ai) £ <~)n<B;).

2.1. Suppose the basic inference was an (~ L)-inference, {t\ R)-inference or (vL)-inference. These have two premises ~[B;] (i=1,2) with <B;)£<B). Therefore «AI /\A2»£<~)n<B). By an (/\ R)-inference and by using Theorem 7.4, from I-~[A;] and I-(Ai~ B;) we obtain I-~[(Al /\A 2)] and I-«Al /\A2)'~ Bi ) (i = 1, 2). In the present case

is also a basic inference. Hence we have I-«Al /\ A2) ~ B). Therefore (AI /\ A2) is an interpolant between ~ and B.

2.2. Suppose the basic inference was an (v R)-inference, (VR)-inference or (3L)-inference. This has one premise ~[Bl]' If the basic inference has an eigenvariable then this does not occur in ~. In every case we have <~)n<Bl) £

<~)n<B). Hence <AI) £ <~)n<B). In the present situation we also have that

is a basic inference. So from I-(Al ~ Bl) we have I-(Al ~ B). Since I-~[Al] also holds, Al is an interpolant between !Yt and B.

2.3. Suppose the basic inference was an (VL)-inference or an (3R)-inference. These have one premise ~[Bl]. Then

is also a basic inference. So from I-(Al ~ Bl) we have I-(Al ~ B). There is a free object variable a such that <Bl ) £ <B)u{a}. Ifaoccursin Bthen <AI) £ <~)n<B). Since we also have I-~[Al]' Al is therefore an interpolant between ~ and B. Now assume that a does not occur in B. Then Al is a formula ffl[a] such that a does not occur in the I-place nominal form fFl . By an (3R)-inference and an (3L)-inference, from 1-~[fFl[a]] and I-(ffl[a] ~ B) we have 1-~[3xfFl[X]] and 1-(3xffl[X] ~ B). Then <3xfFl [x]) £ <~)n<B). Hence 3xfFl [x] is an interpolant between ~ and B.

3. Suppose ~[B] was obtained by a basic inference whose principal part is not contained in B. Then this basic inference is an L-inference.

3.1. Suppose the basic inference was an (~ L)-inference whose secondary part


contains the formula B. This inference has two premises ~[(C1 ~ C2), C1] and ~[C2' 9lo[B]], where &l is the R-form ~[(C1 ~ C2), 9lo]. By I.H. for the second premise there is a formula A such that 1-~[C2' 9lo[A]], I-(A ~ B) and <A) S;

<~[C2' &lo])n<B). Hence <A) s;<9l)n<B) .. By an (~L)-inference from 1-~[(C1 ~ C2), C1] and 1-~[C2' 9lo[A]] we have 1-~[(C1 ~ C2), &lo[A]], that is, I- 9l[A]. Hence A is an interpolant between &l and B.

3.2. Suppose the basic inference was an (~ L)-inference whose secondary term does not contain the formula B. This inference has two premises ~[(C1 ~ C2 ),

9lo[ C 1]] and ~[C2, &lo[Bo]], where B is the formula &lo[Bo] and 9l is the R-form ~[(C1 ~ C2), *1]. Since the first premise ~[(C1 ~ C2), &lo[C1]] is deducible with order <n, by Theorem 7.4 9l0[~[(C1 ~ C2), C1]] is also deducible with order <no By I.H. there is therefore a formula A1 such that l-&lo[A 1], I-(A1 ~ ~[(C1 ~ C2), C1]) and <A1)S;<&lO)n<~[(C1 ~ C2), C1J>. Since the second premise is also deducible with order < n, by I.H. there is also a formula A2 such that 1-~[C2' A2], I-(A2 ~ 9lo[Bo]) and <A2)s; <~[C2' *lJ>n<9lo[BoJ>. Since <~[(C1 ~ C2), C1J>s;<9l), <~[C2' *1])S;<&l) and 9lo[Bo]=B, it follows that «A1 ~ A 2»s; <&l)n<B). By Theorem 7.4 from I-(A1 ~ ~[(C1 ~ C2), C1]) and 1-~[C2' A 2] we have 1-~[(C1 ~ C2), (A1 ~ C1)] and 1-~[C2' (A1 ~ A2)]. Using an (~ L)-inference we have 1-~[(C1 ~ C2), (A1 ~ A2)], that is, 1-9l[(A1 ~ A2)]. By Theorem 7.4 from 1-9l0[A1] we have 1-«A1 ~ A2)~ 9l0[A1])' This together with I-(A2 ~ &lEBo]) by an (~ L)-inference gives I-«A1 ~ A2) ~ 9lo[Bo]), that is, 1-«A1 ~ A 2) ~ B). Hence (A1 ~ A 2) is an interpolant between &l and B.

, 3.3. Suppose the basic inference was an ( v L)-inference. This inference has two premises ~[Ci' B] (i= I, 2) where &l is the R-form ~[(C1 v C2), *1]. By I.H. there are formulas Ai such that I-~[Ci' A;], I-(Ai~ B) and <A;)S; <~[Ci' *lJ>n<B). Hence <CAl v A 2)s; <&l)n<B). By an (v R)-inference from I-~[Ci' AJ we have I-~[Ci' (A1 v A2)] (i= 1,2) and by an (v L)-inference 1-~[(C1 v C2), (A1 v A 2)], that is, 1-9l[(A1 v A 2)]. By an (v L)-inference from I-(Ai~ B) (i=1,2) we have I-«A1 VA2)~ B). Hence (A1 vA 2) is an interpolant between 9l and B.

3.4. Suppose the basic inference was an (V'L)-inference. This inference has a premise ~[(V'xff[x] 1\ ff[a]), B] where &l is the R-form ~[V'xff[x], *1]. By I.H. there is a formula A1 such that I-~[(V'xff[x] 1\ ff[a], A 1]), I-(A1 ~ B) and <A 1) s;<~[(V'xff[x] 1\ ff[a]), *lJ>n<B). Consequently <A1) s;«&l)u {a})n<B). By an (V'L)-inference from 1-~[(V'xff[x]l\ff[a]),A1] we have I-~[V'xff[x], AIl, that is, 1-9l[A1]. If a occurs in 9l, thert <A1) s; <&l)n<B) and hence A1 is an interpolant between &l and B. Now assume that a does not occur in &l. A 1 is a formula ff 1 [a], where a does not occur in ff l' By an (V'R)-inference and by Theorem 7.5 from 1-9l[ff l[a]] and I-(ff 1 [a] ~ B) we have 1-&l[V'Xff1[X]] and 1-(V'Xff1[X] ~ B). Then <V'xff1[xJ> S; <&l)n<B). Hence V'xff1[x] is an interpolant between &l and B.

3.5. Suppose the basic inference was an (3L)-inference. This inference has one premise ~[ff[a], B] where 9l is the R-form ~[3xff[x], *1]. By I.H. there is a formula A such that I-~[ff[a], A], I-(A ~ B) and <A) S; <~[ff[a], *lJ>n<B). The eigenvariable a of the (3L)-inference does not occur in B. Therefore it does not occur in A either. Consequently <A) S; <&l)n<B), and by an (3L)-inference from


I-!,}[ff[a], A] we have 1-!,}[3xff[x], A], that is, 1-91[A]. Hence A is an interpolant between 9l and B.

Remark. The above proof is constructive. I t shows how to construct an interpolant between 9l and B from a given deduction of 9l[B].

2. Interpolation Theorem for the System CP

We are here content with the following simple version:

Interpolation Theorem. For every formula (A -4 C) deducible in CP there is a formula B with thefol/owing properties:

1. (A -4 B) and (B -4 C) are deducible in CPo 2. (B) £(A)n(C).

We call such a formula B an interpolant between A and C.

This theorem can be proved directly in CPo But we can derive it as follows from the interpolation theorem for the system IP3.

By Theorem 8.2 from ~(A -4 C) we have fl(A -4 C). By the interpolation theorem for the system IP3 it follows that there is a formula B such that fl(A -4 B), fl(B-4 C) and (B) £(A)n(C). By Theorem 8.1 wehave~A-4 B)and~B-4 C). By Theorem 8.3 we have ~A -4 A) and ~(C -4 C). By Theorem 4.6 therefore ~A-4 B)and~B-4 C). From (B) £(A)n(C) we have (B) £(A)n(C). Hence B is an interpolant between A and C.

We now give two applications of the interpolation theorem.

3. Finitely Axiomatisable Theories

We consider mathematical theories which can be formulated in the language of the predicate calculus. Such a theory is said to be finitely axiomatisable if it can be based on finitely many formulas of predicate calculus as axioms. We can then take the conjunction of these formulas to obtain a single formula A. The theory ff(A) for which A is the axiom is determined in predicate calculus as follows.

The free object variables, sentential variables and predicate letters occurring in A denote the relevant concepts of the theory ff(A), namely the underlying objects, sentences and predicates which are axiomatically characterized by the formula A.

A formula B expresses a classically (or intuitionistically) provable theorem of the theory ff(A) if the formula (A -4 B) is deducible in the system CP (or IP3).

The theory ff(A) is said to be classically (or intuitionistically) consistent if the formula (A -4 -1) is not deducible in CP (or IP3).

A theory ff(B) is said to be more general than a theory ff(A) if the formula (A -4 B) is deducible. In this case every provable theorem of the theory ff(B) is also provable in the theory ff(A).


The interpolation theorem says: If C is provable in a theory ff(A) then C is also provable in a more general theory ff(B) whose relevant concepts are only those relevant concepts of ff(A) which occur in C. That is: Those relevant concepts of ff(A) which do not occur in C are not needed for the proof of C in ff(A).

This holds not only for theories in classical predicate calculus but also for those in intuitionistic predicate calculus.

4. Beth's Definability Theorem

The simplest version of the definability theorem runs: If the formula

(1) «d'[p] 1\ d'[q]) - Vx(p(x) - q(x)))

is deducible then there is a one-place nominal form §", in which neither p nor q occurs, such that the formulas

(2) (d'[p] - Vx(p(x) - §"[x])) (3) (d'[p] - Vx(§"[x] - p(x)))

are deducible. Here p and q are to be two distinct one-place predicate letters which do not occur in the nominal form d'.

That is to say: If the uniqueness condition (1) is deducible for d'[p] then a ptedicate is defined which can be expressed in predicate calculus by a nominal form §". By (2) and (3) this predicate is uniquely determined by the nominal form d'.

This definability theorem holds not only for classical predicate calculus but also for intuitionistic predicate calculus. Following G. Kreisel it can be proved from the interpolation theorem as follows.

Since the formula (1) is deducible so too is the formula

«d'[p] I\p(a)) - (d'[q] - q(a))).

Here a is a free object variable which does not occur in the nominal form d'. By the interpolation theorem there is therefore a formula A, in which neither p nor q occurs, such that the formulas

«d'[p] I\p(a))- A), (A - (d'[q] - q(a)))

are deducible. We may replace q in the last formula by p. A is a formula§"[a] where a does not occur in §". Therefore we have

(4) f-«d'[p] I\p(a)) - §"[a]), (5) f-(§"[a] - (d'[p] - p(a)).

From (4) it follows that (2) is deducible and from (5) it follows that (3) is deducible. Versions of the definability theorem for many-place predicates can also be

formulated and proved in analogous fashion.

Chapter IV

Classical Simple Type Theory

Like predicate calculus, simple type theory deals with a non-empty domain of basic objects. But it deals not only with predicates over this domain, but also with arbitrary predicates of predicates. Further, quantification is permitted n<;>t only over the basic domain but also over predicates of all types.

We denote the type of the basic objects by 0, the type of statements (formulas) by 1 and the type of predicates with arguments of types 'b ... , '. by ('1' ... , '.). The primitive logical symbols we use are just -4 and V from which we can define 1.., --', 1\, v and 3. We use the abstraction symbol A to form the predicate Ax~I ... x~nd[Xl',···,x~nJ of type ('1' ... ".). This predicate holds for those and only those elements e~', ... , e~n of types '1' ... , '. for which the statement (formula) d[e~', ... , e~nJ holds (is deducible). In addition we use symbols for particular basic objects and for functions on the basic domain.

We first develop classical simple type theory purely syntactically using only constructive methods and then define a corresponding semantics which can only be adequately developed by using non-constructive methods.

§1O. The Formal System CT

1. The Formal Language

1. Inductive definition of the types. 1.1. 0 and 1 are types. 1.2. If'l, ... , '. (n~ 1) are types, then ('b ... , '.) is also a type. We use

a, ai' 'i as syntactic symbols for types and , for sequences, 1, ... , '. (n ~ 1) of types.

2. Primitive symbols of the system CT. 2.1. Denumerably infinitely many free and bound variables of each type. 2.2. A non-empty set of object symbols. 2.3. Certain function letters with specified numbers ~ 1 of arguments. 2.4. The symbols -4, V and A. 2.5. Round brackets and comma.

10. The Formal System CT

For syntactic symbols we use aa, a~i for free variables of types (1, 't;, xa, x? for bound variables of types (1, 't;, at for sequences al', ... , a~n (n ~ I) of free variables, xt for sequences xI', "', x~n (n ~ 1) of pairwise distinct bound variables.

3. Inductive definition of terms in the system CT and their types. 3.1. Every free variable of type (1 is a term of type (1.

3.2 Every object symbol is a term of type O.

57

3.3. If 4> is an n-place function letter (n ~ I) and t?, ... , t~ are terms of type 0, then 4>(t?, ... , t~) is a term of type O.

3.4. Iftl', ... , t~n(n~ I) and t~l,···,tn) are terms of types 't 1, ... ,'to and ('tl' ... , 'to), then tg,,···,tn) (ti', . .. , t~n) is a term of type 1. (We sometimes denote this by tgV) for brevity where 't ='tI' ... , 'to and tt =(ti', ... , t~n.)

3.5. If A and H are terms of type I then (A -+ H) is also a term of type 1. 3.6. If ff[aa] is a term of type I and xa is a bound variable which does not

occur in the nominal form ff then 'v'xaff[x1 is also a term of type 1. 3.7. If d[al', ... , a~n] (n~I) is a term of type 1 and xl', ... ,x~n are pairwise

distinct bound variables \\1!.ich do not occur in the n-place nominal form d then .hi' ... x~nd[xl', ... , x~n] is a term of type ('t I, ... , 'to). (We sometimes denote this by Axtd[xt] for brevity where xt=xl', ... , x~n.)

The terms of type I are called formulas. Free variables of type I and formulas of the form a~ .. ···'tn)(ti', ... , t~n) where a~ .. ···'tn) is a free variable, are called atomic fOrmulas.

For syntactic symbols we use F, tIi for terms of types (1, 't;, tt for sequences ti', ... , t~n (n ~ 1) of terms, A, A;, H, H;, C, C;, F, F;, G, G; for formulas, P, P; for atomic formulas, ff, IF; for I-place nominal forms such that 'v'~ff[xa], 'v'xalF;[xa] are formulas, d, d; for n-place nominal forms (n~ 1) such that Axtd[xt], Axtd;[xt] are terms

of type ('t) where 't='tl' ... , 'to' We identify terms which only differ in the choice of the bound variables

occurring in them. When we write ff[F] or d[tt] we assume that the bound variables which occur in F and tt are chosen so that they do not also occur in the nominal form ff or d (respectively).

Corollaries: 1. if'v'xaff[xa] is a formula and F a term of type (1, then ff[F] is a formula.

2. ifAXtd[xt] isa term of type ('t)= ('t I , ... , 'to) and tt=t1', ... , t~n is a sequence of terms of types 't I"'" 'to then d[tt] is a formula.

2. Chains of Subterms

We need to give formulas a rank for our proofs by induction. We define this rank with the aid of chains of subterms.

58 IV. Classical Simple Type Theory

We call terms of type 0 and free variables prime terms. We define the sub terms of a term as follows: STl. A prime term has no subterms. ST2. An atomic formula a~""" tn)(ti', ... , t~n) has ti', ... , t~n as subterms. ST3. A formula Axtd[xt] (tt) has d[l'] as subterm. ST4. A formula (A -4 B) has A and Bas subterms.

ST5. A formula Vxtr~[xtr] has as subterms every formula ~[atr] (for every free variable if of type 0).

ST6. A term Axtd[xt] has as subterms every formula d[at] (for every sequence at = ai', ... , a~n of free variables of types, 1, ... , 'n)'

By a sub term chain of a term ~ we mean a sequence

toO, ti', t~>' ...

of terms formed as follows: SCI. The initial term toO of the subterm chain is the term ttr. SC2. If a term t~i in the subterm chain is not a prime term, then the subterm

chain contains as immediate successor of t~i a subterm ti+.+ I of tii. SC3. If a term tii in a subterm chain is a prime term then it is the last term in

the subterm chain. We then say that the subterm chain has length i. A term ttr is said to be regular if all subterm chains of ttr have a maximal finite

length m. We define this number m to be the rank Rttr of a regular term ttr. In the following we suppose d is a I-place nominal form such that d[atr ] is a

term.

Lemma 1. If d[ai] is a regular term, then d[a~] is a regular term with the same rank as o[ai].

Proof The lemma follows by induction on the rank of d[ai].

Lemma 2. If every sub term of a term ttr is regular, then ttr is regular.

Proof If a term has infinitely many regular subterms (by ST5 or ST6), then by Lemma I these have the same rank, m say. This yields the assertion since then ttr has rank m + I.

We define the height of a type (J to be the number of round brackets which occur in (J.

Lemma 3. The following conditions are sufficient for a term d[ttr] to be regular: (I) d[a tr ] and ttr are regular terms. (2) For every type 'i of height less than (J we have: If1f'[ai i] and tii are regular

terms, then 1f'[tii] is a regular term. (3) For every regular term di [atr] of smaller rank than d[atr ], d;[ttr] is a regular

term.

10. The Formal System CT 59

Proof l. Suppose 0 is the nominal form * l' Then o[t"] is the term t" which is regular by condition (I).

2. Suppose 0 is a nominal form *1(01' ... , 0.) (n~ I), where 01, ... , o. are I-place nominal forms. Then O"=('b ... , '.) where 0i[t"] is of type 'i. For i= I, ... , n, oJa"] is a regular term of smaller rank than o[a"]. It follows by condition (3) that oJt"] is regular.

2.l. Suppose t" is a prime term. Then o[t"] has the subterms 0l[t"], ... , o.[t"]. Since these are regular, o[t"] is regular.

2.2. Suppose t" is a term .A.xtd[xt] where '='1' ... , ' •. Then t" has a subterm d[at] where at =a~', ... , a~n. This subterm is regular since t" is regular by con-dition (l). For i= I, ... , n, 'i has a smaller height than 0". Since oJt"] is regular it follows by condition (2) that d[ol[t"], ... , o.[t"]] is regular. This is the only sub term of o[t"]. Hence o[t"] is regular.

3. In every other case every subterm of o[t"] is of the form oJt"], where 0i[a"] is a subterm of o[a"]. Since 0i[a"] is then a regular term of smaller rank than o[a"], it follows by condition (3) that every subterm 0i [t"] of o[t"] is regular. Hence o[t"] is also regular.

Lemma 4. If o[a"] and t" are regular terms then o[t"] is also a regular term.

Proof It follows from Lemma 3 by induction on the rank of the regular term o[a"] that o[t"] is regular if conditions (l) and (2) of Lemma 3 hold. The assertion now f?llowS by induction on the height of the type 0".

We define the length of a term t" to be the number of primitive symbols which occur in t".

Theorem 10.1. Every term t" is regular.

Proofby induction on the length of the term t". l. Suppose t" is a formula .A.xtd[xt] (tt) where tt = t'1', ... , t~n. Then the terms

d[at] and t'1', ... , t~n have smaller lengths than t". By the induction hypothesis these are therefore regular. By Lemma 4 it follows that the term d[tt] is also regular. This is the only subterm of t". Hence t" is regular.

2. In every other case every subterm of t" has a smaller length than t" and therefore by the induction hypothesis is regular. Hence t" is regular by Lemma 2.

Corollary to Theorem 10.1: Every formula has a computablejinite rank.

Theorem 10.2. Thefollowing holdfor ranks offormulas: (1) RA<R(A-4B) and RB<R(A-4B). (2) Rff[a"] < RVx"ff[x"]. (3) Rd[t'] < Rhtd[xt](tt).

Proof Immediate from the definition of the rank of a formula and the definition of subterm of a term.


3. Axioms and Basic Interferences

We set

and define P-forms, N-forms, NP-forms, positive parts, negative parts, minimal positive and negative parts and

as in §3. As before we use the following syntactic symbols:

f!J, f!Ji for P-forms, %, %i for N-forms, 8, 8 i for P-forms and N-forms, fl, fli for NP-forms.

The axioms of the system CT are all formulas of the form fl[P, P].

The basic inferences of the system CT are: (Sl) %[(A - 'v'X 1X1)], %[B] I- %[(A - B)],

if B is not a formula 'v'X 1X1. (S2) f!J[,F[aa]] I- f!J['v'xa ,F[xa]] (with a condition on the variable). (S3) (,F[ta] _ %['v'xa,F[xa]]) I- %['v'xa,F[xa]]. (S4) 0"[d[tt]] I- S[htd[xt](t')].

The free variable denoted by aa in the premise of an (S2)-inference is called the eigenvariable of the inference. It must not occur in the conclusion. The minimal positive or negative part in the conclusion of a basic inference is said to be the principal part of the given basic inference.

4. Deducible Formulas and Permissible Inferences

Theorem 10.3. Every formula of the form fl[ C, C] is deducible.

Proofby induction on the rank of the formula C in the same way as for Theorem 4.1. For the case that C is a formula Axtd[xt](tt) we start from the assumption that, by I.H., fl[d[tt], d[tt]] is deducible. Using (S4)-inferences we have I- fl[C, C].

Theorem 10.4. Every formula %['v'X1X1] is deducible.

Proof % is an N-form f!J[(*1 - B)]. By Theorem 10.3 the formula (Bf!J[('v'X 1X1 _ B)]) is deducible. Using an (S3)-inferencewe have I- f!J[('v'X1X1 - B)], that is, I- %['v'X1X1].

Coronary to Theorems 10.3 and 10.4. The formal system CT is sententially complete.

10. The Formal System CT 61

Proof The sentential completeness follows from Theorem 10.4 in the same way as for the system CP (see §4.1) since the connective 1.. is defined by V'X 1X 1 and hence P-forms and N-forms are defined as in CPo

Theorem 10.5. (Particular substitution rule). If a~ does not occur in the nominal form fF, then


Proof Like that for Theorem 4.2.

Theorem 10.6 (Inversion rules). The following are weak inferences. (1) %[(A -> B)] f- %[(A -> V'X 1X 1)].

(2) %[(A -> B)] f- %[B]. (3) .o/I[V'xtIfF[xtI ]] f- .o/I[fF[atI ]]. (4) fF[A.xtd[xt] (tt)] f- fF[d[tt]].


Theorem 10.7 (Structural rule of inference). If F~ G holds, then F f- G is a weak inference.


Theorem 10.8. %[fF[ttI]] f- %[V'xtIfF[xtI]] is a permissible inference.


Theorem 10.9 (General substitution rule). If if does not occur in the nominalform fF, then

is a permissible inference.

Proofby induction on the deduction. 1. Suppose fF[d'] is an axiom. Then fF[n is a formula of the form ,q[C, C],

which is deducible by Theorem 10.3. 2. Suppose F[d'] was obtained by a basic inference. Then by Theorem 10.5

we can assume that the basic inference does not have any free variable occurring in r as its eigenvariable. Then the assertion follows from the induction hypothesis.

The logical connectives, 1\, v and 3 can be defined as in the system CP and for these one obtains the permissible inferences as given in Theorem 4.8.


5. The Cut Rule

While hitherto all the proofs that certain inferences are permissible have been constructive, the cut rule

gJ[A], (A - B) f- gJ[B]

cannot be shown to be permissible for the system CT by using constructive methods. The permissibility of the cut rule in the system CT corresponds to the funda

mental conjecture from which G. Takeuti [IJ has constructively proved the consistency of a formal system of classical analysis.

But by K. Godel [1] the consistency of a formal system of classical analysis cannot be proved by constructive methods. Hence the permissibility of the cut rule in CT is not constructively provable.

The permissibility of the cut rule was first proved by W. Tait [lJ for s~cond order predicate calculus (a subsystem of classical simple type theory) and then by M. Takahashi [1 J and D. Prawitz [lJ for the full classical simple type theory. These proofs are related to a semantic equivalent of the syntactic fundamental conjecture of Takeuti which was developed by K. Schutte [4J using partial valuations.

In §12 we introduce suitable semantics and prove the completeness theorem using the methods of Takahashi and Prawitz. From this it follows that the cut rule is permissible. Before doing this we establish in §11 the relation between deducibility and the existence of partial valuations which we need for the completeness proof.

§11. Deduction Chains and Partial Valuations

1. Definition of Deduction Chains

We define deduction chains in the same way as for the system CP (see §5.3). We first assume that for each type (J an enumeration

ag, a~, a~, ...

of all free variables of type (J and an enumeration

of all terms of type (J are given. The reducible parts of a formula F are defined to be the positive parts of F

which are oftheform Vx" ff[x"J or Axtd[xtJ (tt) and the negative parts of Fwhich are of the form Axtd[xtJ(t') or (A- B) where B is not a formula VX1Xl. A formula is said to be reducible if it has at least one reducible part. By the distinguished

11. Deduction Chains and Partial Valuations 63

part of a reducible formula Fwe mean the reducible part of Fwhich occurs furthest to the right in F.

By the critical parts of a formula Fwe mean the negative parts of Fwhich are of the form VxtJ 3"[xtJ ]. A formula is said to be critical if it is not reducible and has at least one critical part.

A formula is said to be primitive if it is neither reducible nor critical. This is obviously the case if, and only if, every minimal positive part and every minimal negative part of the formula is an atomic formula.

By a D-chain (deduction chain) of aformula Fwe mean a sequence offqrmulas

formed as follows: DI. The initial formulaFo of the D-chain is the formula F. D. Ifa formula Fm of the D-chain is an axiom or an atomic formula, then it is

the last formula of the D-chain. We then say that the D-chain has length m. D3. If a formula Fm of the D-chain is a reducible or critical formula but not an

axiom, then Fm has an immediate successor Fm+l in the D-chain, where Fm+l is determined by Fm as follows:

D3.I. If Fm is a reducible formula %[(A - B)] with distinguished negative part (A - B), then Fm+ 1 is either the formula %[(A - VX 1 X 1)] or the formula %[B].

D3.2. If Fm is a reducible formula &'[VxtJ3"[xtJ]] with distinguished positive part VxtJ3"[xtJ ], then Fm+ 1 is the formula &'[3"[o:iJ] where i is the least number such that o:i does not occur in Fm.

D3.3. If Fm is a reducible formula C[htd[xt](tt)] with distinguished positive or negative part Axtd[xt] (tt) then Fm+l is the formula C[d[tt]].

D3.4. If Fm is a critical formula with critical parts

then Fm+ 1 is the formula

where Ai+ j.k: = 3";[sj'] (i = 1, ... , k; j = 0, ... , m).

2. Partial Valuations

By a partial valuation we mean a function V which assigns a truth value VF; = t or VF; =fto certain formulas F; (but which may not be defined for all formulas) such that the following conditions are satisfied:

VI. If V(A - B)= t, then VA =f or VB=t. (In this case VB or VA may be undefined.)

V2. If V(A-B)=j, then VA=tand VB=!


V3. If V'v'x" 3i'[x"] = t, then V3i'[t"] = t for every term t" of type (J.

V4. If V'v'x"3i'[x"]=f, then there is a free variable a" of type (J such that V3i'[a"] = f (In this case V3i'[t"] need not be defined for every term t" of type (J.)

V5. If VAx"d[x"] (t") is defined then V d[t"] = Vh"d[x"] (t").

Remark. These partial valuations correspond to the semivaluations in Schutte [4], Takahashi [1] and Prawitz [1].

3. Principal Lemmata

As in §5.3 we have:

Principal Syntactic Lemma. If every D-chain of a formula F contains an axiom then the formula F is deducible.

Proof In this case every D-chain of F is finite. Hence by Konig's lemma Fhas only finitely many D-chains. These constitute a deduction of the formula F.

Principal Semantic Lemma. If there is a D-chain of formula F which contains no axiom then there is a partial valuation V such that VF=f

Proof Suppose

(DC) Fo, F l , F2''''

is a D-chain which contains no axiom where Fo is the formula F. We now prove, in analogy with §5.3:

Lemma 1. If an atomic formula occurs as a positive (or negative) part in aformula Fn of the D-chain, then it also occurs as a positive (or negative) part in every formula Fm (m ~n) of the D-chain.

Proof This follows immediately from the definition of D-chain.

Lemma 2. If a formula Fn of the D-chain has a reducible positive (or reducible negative) part C, then the D-chain contains a reducible formula Fm (m ~n) with distinguished positive (or negative) part C.

Proof By induction on the sum of the ranks of the reducible parts of Fn which occur in Fn to the right of C.

Lemma 3. If a formula Fn of the D-chain has a negative part 'v'x"3i'[x"] then the D-chain contains infinitely many critical formulas with negative part 'v'x"3i'[x"].

Proof Under the hypothesis for Lemma 3 it suffices to prove that the D-chain contains a critical formula Fm (m > n) with negative part 'v'x" 3i'[x"]. We call the sum of the ranks of the reducible parts of a formula the reducibility rank of the

12. Semantics 65

formula. By hypothesis the formula Fn is either reducible or critical. The D-chain therefore contains a formula Fn+ 1 which also contains the formula Vxa§"[xa] as a negative part. We prove our assertion by induction on the reducibility rank of the formula Fn+ l' If Fn+ 1 is a critical formula then the assertion is satisfied with m=n+ 1. Otherwise the D-chain contains a formula F,,+2 which has a smaller reducibility rank than Fn+ 1 and also contains the formula Vxa§"[xa] as a negative part. In this case the assertion follows from the induction hypothesis.

Now we set VC= t (or VC=f) if C occurs in the D-chain (DC) as a negative (or positive) part.

Lemma 4. Thefunction V satisfies conditionsVI-V5 for a partial valuation.

Proof 1. Suppose V(A --+ B) = t, then (A --+ B) is a negative part in (DC). If B is a formula VX 1X1, then A is a positive part in (DC) and therefore VA =f Otherwise it follows from Lemma 2 and the definition of D-chain that A is a positive or B is a negative part in (DC), and therefore VA = for VB= t.

2. Suppose V(A --+ B) = j, then (A --+ B) is a positive part in (DC). But then A is a negative and B a positive part in (DC), so VA = t and VB= f

3. Suppose V'r/xa §"[xa] = t, then Vxa §"[xa] is a negative part in (DC). From Lemma 3 and the definition of D-chain it follows that, for every term ta of type (1,

§"[ta] is a negative part in (DC). Hence V§"[ta] = t for all ~ of type (1.

4. Suppose V'r/xa§"[xa]=j, then Vxa§"[xa] is a positive part in (DC). From Lemma 2 and the definition ofD-chain it follows that there is a variable aa of type (1

such that §"[aa] is a positive part in (DC), and therefore V§"[aa] = f 5. Suppose VAxtd[xt] (tt) is defined. Then Axtd[xt](tt) is a positive (or

negative) part in (DC). From Lemma 2 and the definition ofD-chain it follows that d[tt] is a positive (or negative) part in (DC). Hence V d[tt] = VAxtd[xt] (tt).

Since the D-chain (DC) contains no axiom it follows from Lemma 1 that no atomic formula occurs both as a positive and as a negative part in (DC). Therefore there is no atomic formula P for which VP = t and VP = f From Lemma 4 it follows by induction on the rank of the formula C that there is no formula C for which VC = t and VC = f Hence V is a partial valuation. Hence VF = j, since F occurs as a positive part in (DC).

§12. Semantics

1. Total Valuations over a System of Sets

By a system M of sets we mean a collection of non -empty sets Ma for each type (1,

such that Mal and Ma 2 are disjoint if (11 #- (12'

For a given system M of sets we introduce the following terminology. The basic M-terms of type (1 are the names ofthe elements of M a. The M-terms of type (1 are the expressions which result from terms of type (1


when all the free variables occurring in them are replaced by basic M -terms of the same type.

The M-formulas are the M-terms of type 1. By an M-variant of a formula Fwe mean an M-formula which is obtained from

Fby replacing the free variables occurring in Fby M-terms of the same type where a free variable which occurs more than once in F is replaced at all occurrences by the same M-term.

As syntactic symbols we use c", cii for basic M-terms of type CT, rio c< for sequences ell, ... , c~" (n ~ 1) of basic M-terms, utI, Uii for M-terms of type CT, rio u< for sequences ul l , ••• , U~" (n~ 1) of M-terms, A', B', C', F', G', 'Ix"ffTx"] for M-formulas and Ax<dTx<] for M-terms of type (r) = (rl' ... , tft)'

Bya total valuation over a system M of sets we mean a function Wwhich assigns a truth value WF' = t or WF' = fto each M-formula F' such that the following hold:

WI. W(A'-B)=tif,andonlyif, WA'=for WB'=t. W2. W'Ix"ffTx"]=t if, and only if, Wff'[u"]=t for every M-term utI of

type CT.

W3. WAx<dTx<] (u<)= WdTu<]. A formula Fis said to be true in a total valuation Wover M if WF' = t for every

M-variant F' of F. A formula is said to be valid if it is true in every total valuation over a system

of sets.

Remark. An adequate semantics for the formal system CT is provided by total valuations but not by set-theoretic models since there is no extensionality axiom in CT. The semantics given here are therefore weaker than the semantics given by general models which L. Henkin [1] showed is adequate for classical simple type theory with extensionality.

2. Soundness Theorem

We denote by f!IJ', %' and!2' the nominal forms obtained from P-forms, N-forms and NP-forms when the free variables in them are replaced by basic M-terms of the same types.

For every total valuation Wover M we obviously have:

PN-Lemma. If WA'=t (or WA'=f), then Wf!lJTA']=t(or W%TA']=t).

Theorem 12.1 (Soundness Theorem). Every deducible formula is valid.

Proof Let F be a deducible formula, Wa total valuation over a system M of sets and F' an M-variant of F. We prove that WF' = t by induction on the order of a deduction of F.

12. Semantics 67

1. Suppose F is an axiom ~[P, P]. Then F' is an M-formula ~'[A', A']. By the PN-Iemma we have WF'=t.

2. Suppose F is a formula %[(A -4 B)] obtained by an (SI)-inference from %[(A -4 Vx l Xl)] and %[B]. Then F' is an M-formula %'[(A' -4 8)]. If W A' = t, then W(A' -48)= WB'. Since, by I.H., W%'[B'] =t, we have WF'=t. If WA'=J, then W(A'-4 B')=t= W(A'-4 VX1Xl). Since, by I.H., W%'[(A'-4 VX1Xl)]=t, we have WF' = t.

3. Suppose F is a formula &I[Vxa ff[xa]] obtained by an (S2)-inference from &I[ff[aa]]. Then F' is an M-formula &I' [Vxa F'[xa]]. If WVxa ff'[xa] = t, then WP = t by the PN-Iemma. If WVxaff,[xa] = J, then there is an M-term ua such that Wff'[ua] =/ &I'[ff'[ua]] is an M-variant of &,[ff[aa]]. Hence, by I.H., W&I,[ff'[ua]]=t. Therefore WF'=t.

4. Suppose Fis a formula %[Vxaff[xa]] obtained by an (S3)-inference from (ff[ta] -4 %[Vxaff[xa]]). Then F' is an M-formula %'[Vxaff'[xa]]. By I.H., we have W(ff'[ua] -4 %'[Vxa ff'[xa]]) = t for an M-term ua. If Wff'[ua] = t, then WF' =t.IfWff'[ua] =J,then WVxaff'[xa] =/too. Then WF' =tbythePN-lemma.

5. Suppose F is a formula 1[A.x'd[x'](t')] obtained by an (S4)-inference from 8[d[t']]. Then F' is an M-formula 8'[AXd'[x'](u')]. By I.H., W8'[d'[u']] =t. Since WAx'd'[x'](u') = Wd'[u'] it follows that WF' =t.

3. Extending a Partial Valuation

I:.et V be a partial valuation. We shall define a total valuation Wover a suitable system M of sets such that no formula F for which VF=/ is true in the total valuation W. We follow D. Prawitz [1].

Inductive definition of a set pta (of possible valuations) for each term ta (by induction on the height (see p. 58) of the type 0").

PI. PtO :={to}. P2. If Vt l is defined, let Pt l : = {Vt l }. Otherwise let Pt l : = {t,J}. P3. For "t"="t"l' ... ,"t"n (n~ 1) Po E PtS) if, and only if, the following conditions

are satisfied: P3.I. Po is a set of n-tuples

such that Pi E Ptj' (i= 1, ... , n). P3.2. If Vt~)(tl', ... , t:") is defined and Pi EPtj' (i= 1, ... , n), then

if, and only if, Vtb')(tl', ... , t:") = t.

For every type 0" let M a be the set of ordered pairs (ta,p) with P E pta. Obviously Ma is non-empty. So the sets M a form a syste~ M of sets. We now consider this system M and use the syntactic symbols introduced in §I2.1.


If utI is an M-term of type ulet utI. be the term of type u obtained from utI by replacing each basic M-term (tj',p;) by the term t?

Inductive definition of the degree du" of an M-term utI. dl. du": =0 if u=O or utI is a basic M-term. d2. du~)(ul', ... , U~"): =max (du~), dull, ... , du~")+ 1. d3. d(A' --+ B') : =max (dA', dB') + 1. d4. dVx"~'[x"]: =dff'[e"] + 1, where e" is a basic M-term. dS. dAX'd'[x']: =dd'[e'] + 1, where e' is a sequence of basic M-terms.

Inductive definition of Wu" for each M-term utI (by induction on the degree du"). WI. Wuo:=uo*. W2. W(t",p):=p for PEPt". (If u=O, we have p=tO, hence W(tO,p)=

to= (to, p)* by W2, in accordance with WI.) W3. Wu~)(ul', ... , U~") : = t, if

Otherwise let Wu~)(ul', ... ,U~"):=! W4. W(A' --+ B') :=t, if WA'=/or WB'=t. Otherwise let W(A' --+ B'):=! WS. WVx"~'[x"]:=t, if W~'[e"]=t holds for every basic M-term e" of

type u. Otherwise let WVx"~'[x"] :=! W6. For t=tl' ... , til (n ~ I) let WAx'd,[x'] be the set ofn-tuples (eI', ... , e:")

satisfying the condition W d'[ei', ... , e:"] = t.

Proo/by induction on the degree du". 1. Supposeu=O. Then we have Wuo=uo·and Puo·={uo.}, so WUoEPUo •. 2. Suppose utI is a basic M-term (t",p). Then WU"=PEPt" and u"o=I", so

WU"EPU"·. Suppose u= I and u1 is not a basic M-term. By definition Wu 1 E {t,f}. We

now only need to prove that: If Vu 1• is defined, then Wu 1 = Vu 1 •. 3.1. Suppose u1 is an M-formula ul»(ui', ... , U~"). Then by I.H. Wul»EPul)l"

and WuI' E pur (i= 1, ... , n). If Vu1• is defined then, by the definition of Pu~l",

if, and only if, Vu1• '= t. Hence in this case Wu1 = Vu1 •. 3.2. Suppose u1 is an M-formula (A' --+ B'). Then by I.H. WA' EPA'· and

WB' EPB'·. If Vu1·=t, then VA'·=/or VB'· =t. Then by I.H. WA'=/or WB'=t and therefore Wu1=t. If Vu1• =J, then VA'·=I and VB'·=! Then by I.H. WA'=t and WB'=/and therefore Wu 1=!

3.3. Suppose u1 is an M-formula Vx"o~'[x"O]. Then uh is a formula Vx"O~[x"o]. By I.H. W~'[c"°] E P~[c"°·] for every basic M-term of type uo. If Vu1·=t then V~[t"]=t holds for every term ["0 of type uo. Then by I.H.

12. Semantics 69

W~'[c"°] = t for every basic M-term c"0 of type (Jo and therefore Wu1 = t. If Vu 1" =f, then there is a free variable if ° of type (Jo such that V~[ifO]=f Then by I.H. W~'[c"O]=ffor some basic M-term c"o of type (Jo and therefore Wu1=f

4. Suppose if is an M-terrn htd'[x'] where (J=(-')=(-'I,"" -'n) (n~ 1). Then if" is a term Axtd[xt]. If Vif"(cl l ", ••• , c~n") is defined then V d[cliO , ••• , c~n"] = Vif"(cllO , ••• , c~n.). Then by I.H. W d,[cl l , ••• , c~n] = V d[cl l ., ••• , c;"]. In this case by the definition of Wif

(cl l , ••• , c~n)E Wif

if, and only if, VifO(cII ., ••• , c~n")=t. Hence we have Wu" E Pu"" by the definition of Pif·.

Lemma 2. If 0 is a I-place nominal form such that o[if] is an M-term, then Wo[if] = Woe (if·, Wif)].

Proof By Lemma 1 Wif E PifO and therefore (if", Wif) is a basic M-term of type (J. If 0=*1 then the assertion holds by the definition of W(if·, Wif). In the other cases the assertion follows by induction on the length of the nominal form o.

Lemma 3. W\fx"~'[x"]=t if, and only if, W~'[if]=tfor every M-term if of type (J .

. Proof If W~'[u"]=t for every M-term u" of type (J then W\fx"~'[x"]=t by definition. If W\fx"~'[x"]=t, then W~'[(u"', Wu")]=t and by Lemma 2 W~'[u"] = t for every M-term if of type (J.

Lemma 4. Whtd'[xt](ut )= Wd'[ut ].

Proof Let c~;: = (u;;·, WuD (i = 1, ... , n) and ct : = cI\"" c~n. By definition WAx'd'[xt](ut)=t if, and only if, (cll , ••• , c~n) E Whtd'[x']. This is the case if, and only if, Wd'[ct]=(and then by Lemma 2 Wd'[ut]=t.

Theorem 12.2. The restriction of W to M-formulas is a total valuation over M under which no formula F is true for which VF= f

Proof By the definition of Wand Lemmata 3 and 4 W is a total valuation over M. Suppose VF=f Replace each free variable a occurring in F by a basic M-term ( .. , p) such that p EPa" to obtain an M-formula F'. Then F' is an M-variant of F with P'"=F. By Lemma 1, WF' E PF. Since VF=j, we have WF'=f HenceFisnot true under the total valuation W.

4. Completeness Theorem and Cut Rule

Theorem 12.3 (Completeness Theorem). Every validformula is deducible.


Indirect proof Suppose F is not deducible. Then by the principal syntactic lemma (§11.3) there is a D-chain of F which contains no axiom and therefore by the principal semantic lemma (§11.3) there is a partial valuation V with VF=f It follows from Theorem 12.2 that F is not valid.

Theorem 12.4 (Cut rule). &I[A], (A -+ B) I- &I[B] is a permissible inference.

Proof Suppose the formulas &I[A] and (A -+ B) are deducible, then they are valid by Theorem 12.1. Let Wbea total valuation over a system M of sets. If&l'[A'] and (A' -+ 8) are M-variants of the formulas &I[A] and (A -+ B) then W&I'[A'] = t and W(A'-+B')=t. If WB'=t, thenW&I'[B']=t by the PN-lemma (p. 66). If W8 = f, then since W(A' -+ 8) = t, we have WA' = f Then from W&I'[A'] = t we have W&I'[B']=t. In any case, therefore, W&I'[B']=t. Hence &I[B] is valid and, by Theorem 12.3, deducible. .

Part B

Systems of Arithmetic

Chapter V

Ordinal Numbers and Ordinal Terms

In this chapter we develop a constructive system of ordinals which we shall use in § 16 and Chapter VIII for the proof-theoretic treatment of pure number theory and predicative analysis. In §13 we start from a non-constructive presentation of the classical theory of ordinals in which we take as basis a corresponding axiomatic characterization of the set 0, of all finite and denumerably infinite ordinals where, however, we do not go very far into set theory but use sets in a naive way. In this non-constructive framework we develop a hierarchy of normal functions and critical ordinals which determine a certain initital segment of the ordinals (in fact up to r 0)' This initial segment of the ordinals will be constructively characterized in §14 (by ordinal terms) and all the properties of this system of ordinal torms which we need later will be developed constructively. The full system of these ordinal terms (up to r 0) is first used in Chapter VIII (for predicative analysis) while a subsystem (up to eo) occurs in §16 (for pure number theory).

§13. Theory of Ordinals of the 1st and 2nd Number Classes

1. Order Types of Well-Ordered Sets

Definition. A set M is said to be orderedby a relation R, if, for all a, b, C E M we have: 1. Not a R a (Irreflexiveness) 2. aRb and b R C imply aRc (Transitivity) 3. aRb or a=b or bRa (Connection)

Corollary of 1-3: If a, b E M then preCisely one of aRb, a = b, bRa holds.

M is said to be well-ordered (by R) if we also have: 4. Every non-empty subset M 0 ~ M contains a first element with respect to R.

That is: there exists ao E M 0 such that ao = x or ao R x for all x E Mo. We denote a set M, well-ordered by R, by (M, R). If a E M we let (M, R) I a be

the set of those x E M such that x R a with the relation R restricted to this set. (M, R) I a is again a well-ordered set.

74 V. Ordinal Numbers and Ordinal Terms

Two well-ordered sets (Mt, R t ), (M2' R 2) are said to be isotonic if there is an order preserving map of M t onto M 2, that is, a map f:Mt -> M2 such that j{M t )=M2 and

aRt b=f(a) R2f(b)

for a, bE Mi. Such a map is bijective. Isotonism is an equivalence relation. The equivalence classes of well-ordered sets under isotonism are called order types.

We set (Mt, R t ) «M2' R 2) if there is a2 E M2 such that (Mt, R t ) and (M2' R 2) I a2 are isotonic. This relation is compatible with isotonism. That is: if (Mt, R t ),(M2, R2)areisotonicand(M3 , R 3 ), (M4, R4) are isotonic then (Mt , Rd< (M3 , R 3 ) implies (M2' R2) < (M4' R4)' We can therefore make the following definition for order types IXt, 1X2 : IX t <1X2 if, and only if, (Mt, R t )«M2 , R 2 ) for wellordered sets (Mt, R t ), (M2 , R 2 ) of order types IXt, 1X2' respectively. The totality of order types of well-ordered sets is then well-ordered by this relation <.

Ordinal numbers (or ordinals) serve as representatives of order types of wellordered sets. In the context of axiomatic set theory (say Zermelo-Fraenkel or Bernays-G6del) they are defined as specific sets which are well-ordered by the E-relation in such a way that every order type of a well-ordered set has precisely one ordinal number belonging to it. The totality of these ordinal numbers is wellordered by the E-relation which corresponds to the < -relation on order types here.

The ordinal numbers which belong to order types of finite well-ordered sets form the I st Cantor number class. These ordinals correspond to the natural numbers. The ordinals which belong to order types of denumerably infinite wellordered sets form the 2nd Cantor number class.

2. Axiomatic Characterization of the 1st and 2nd Number Classes

In this section we restrict our attention to the ordinals of the 1 st and 2nd number classes and we represent these by a set 0 which is characterized as a set of ordinals by the three axioms below. These axioms uniquely determine, up to isomorphism, the set of ordinals of the 1 st and 2nd number classes as it arises in the context of an axiomatic set theory. Here we are sticking to an axiomatic determination of the concept of set, in general in what follows we shall be dealing with a naive concept of set.

Ax. I. 0 is a set well-ordered by a relation <.

Ax. II. Every bounded subset of 0 is denumerable. That is: if, given Me 0 there exists IX E 0 such that ~ < IX for all ~ E M, then M is a finite or denumerably infinite set.

Ax. III. Every denumerable subset of 0 is bounded. That is: for each finite or denumerably infinite set Me 0 there exists IX E 0 such that ~ < IX for all ~ E M.

Corollary: 0 is an infinite, but not denumerable set.

13. Theory of Ordinals of the 1st and 2nd Number Classes 75

We call the elements of 0 ordinal numbers (ordinals, for short). We denote them by small Greek letters. We write rx ~ 13 for rx < 13 or rx = 13, rx > 13 for 13 < rx and rx ~ 13 for 13 ~ rx. All the definitions introduced in this section refer to the axiomatically given set O.

If d(rx) is a statement about an arbitrary ordinal rx then the property d of ordinals is said to be progressive if, for all ordinals rx, d(rx) can be deduced from the hypothesis that d(~) holds for all ~ <rx.

Induction Theorem. If d is progressive then d(rx) holds for all ordinals rx.

Proof Suppose d(rx) did not hold for every ordinal rx. Then the set of ordinals ~ such that d(~) is false is non-empty. By Ax. I this set must contain a least element. This contradicts the hypothesis that d is progressive.

The induction theorem shows that in order to prove a theorem d(rx) about ordinals it is sufficient to prove d is progressive. That is: it is sufficient to prove d(rx) for arbitrary rx EO under the

Induction hypothesis: d(~) holds for all ~<rx. This proof principle for d(rx) is called transfinite induction (on rx). We shall

frequently apply this transfinite induction below to prove theorems about ordinals of the set 0.

3. Zero, Successor and Limit Numbers and Supremum

By Ax. I there is a unique smallest ordinal. We denote this by O.

For each ordinal rx there is a unique smallest ordinal> rx.

Proof {rx} is a denumerable subset of 0. By Ax. III it follows that the set of ~ > rx is non-empty. By Ax. I this set contains a unique smallest element. We call this the successor rx' of rx.

The following hold for this definition of rx':

rx < rx' rx<P ~ rx'~p rx<P' ~ rx~p.

An ordinal 13 is said to be a successor number if there is an rx EO with rx' = p. An ordinal is said to be a limit number if it is neither 0 nor a successor number.

For every limit number 13 the following holds:

rx <13 ~ rx' <13.

Proof rx < 13 implies rx' ~ p. If 13 is a limit number then rx' #- 13, hence rx' < p. For every denumerable set Me 0 there is a unique smallest ordinal rx such that

~ ~ rxfor all ~ E M. (This follows from Ax. III and Ax. I.) We call this ordinal rx the

76 v. Ordinal Numbers and Ordinal Terms

supremum sup M of the set M. The definition yields

~ EM=> ~ ::::; sup M ~ ::::; P for all ~ EM=> sup M::::; P P<sup M => there exists ~ E M with P<~.

Theorem 13.1. If M is a non-empty denumerable subset of 0 which has no maximal element then sup M is a limit number.

Proof Since M has no maximal element, sup M If M. For all ~ EMit follows that ~ < sup M. Since M is non-empty it follows that sup M =f. O. If rx'::::; sup M then rx < sup M. Then there exists e E M such that rx <~. It follows that rx'::::; ~ < sup M. Therefore sup M is neither 0 nor a successor number, and hence is a limit number.

Definition. The set N ofjinite ordinals is the smallest subset of 0 which contains 0 and is such that if rx is in the subset then so is its successor rx'.

This set N which consists of the ordinals 0, 0', 0", ... , corresponds to the set of natural numbers. Accordingly we set I : =0', 2: =0", .... Obviously N is a denumerably infinite set which contains no maximal element and no limit number.

Lemma 1. rx<P, PE N => rxE N.

Proof by transfinite induction on p. If rx < P and PEN then it follows that there exists Po EN with p~=p. Then rx::::;Po. If rx=Po, then rx EN. If rx<Po then rx EN by the induction hypothesis since Po < p.

Definition. w: = sup N.

Theorem 13.2. w is the smallest number and rx < w holds if, and only if, rx EN.

Proof By Theorem 13.2 w is a limit number since N contains no maximal element. For all rx E N it follows that rx < w. Conversely, if rx < w, then there exists PEN with rx < p. By Lemma I, rx EN. Hence rx < w if, and only if, rx EN. It follows from this that w is the smallest limit number, since N contains no limit number.

4. Ordering Functions

For rx E 0 let O( rx) be the set of all ~ < rx. 0(0) is the empty set, and by Theorem 13.2 O(w) = N. Every set O(rx) is denumerable by Ax. II.

Definition. A set A ~ 0 is said to be an O-segment if

The whole set 0 is an O-segment. An O-segment which is =f. 0 is said to be a


proper IO-segment. Obviously every lO(a) is a proper IO-segment. The converse also holds:

Every proper IO-segment is an lO(a).

Proof If A # 10 then there is a least ordinal a ¢ A. If A is an IO-segment then A = IO( a).

Definition. A function f: A ----> B is said to be an ordering function of a set B £ 10 if the following holds:

1. The domain A of the functionfis an IO-segment. 2. The function f is strictly monotone, that is, for all ~ < 17 E A we have

f{~)<f{17)· 3. The image of the functionfis the set B.

Remark. If B is empty then the empty function 0 ----> 0 is an ordering function of B. (The empty set 0 is an IO-segment.)

Theorem 13.3. Iff: A ----> B is an ordering function of B then a~a)for alia E A.

Proofby transfinite induction on a. For all ~<a,f{~)<f{a) and by the induction hypothesis ~ ~f{~), therefore ~ <f{a). Hence a ~j(a).

Lemma 2. Every set B£ 10 has at most one ordering function.

Proof Let;;: Ai ----> B (i = l,2) be ordering functions of B. We prove by transfinite induction on a: If a E A l , then a E A2 andfl(a) = f2(a). Now a E Al implies ~ E A 1

for all ~ < a since A 1 is an IO-segment. By the induction hypothesis it follows that ~ E A2 andfl(~) =f2(~) for all ~ <a. By property 3 it follows that a E A 2. By properties 1 and 2 it follows thatfl(a) is the least element ofB not equal to any fl(~) for ~<a and similarly for f2(a). Hence fl(a)=f2(a). Therefore A l £A2. Similarly A2 £ A l · It follows that Al =A2 andfl(a) =f2(a) for all a E A 1 and thereforefl =f2'

Definition. By a proper segment of a set BelO we mean a set of the form B(f3): = BnlO(f3) for some f3 E B.

Lemma 3. If every proper segment of a set Be 10 has an ordering function then B also has an ordering function.

Proof Letfp: Ap ----> B(f3) be an ordering function of B(f3) for each proper segment B(f3) of B. By Ax. II the set B(f3) is denumerable. Since fp is bijective, Ap is also denumerable and therefore a proper IO-segment. Therefore to each ordinal fJ E B there is an ordinal g(f3) such that Ap = lO(g(f3». 9 is a map from B into 10.

(l) If f31' f32 E Band f31 < f32 then g(f3 d <g(f32)'

Proof f p2 is a map of lO(g(f32» onto B(f32)' Since f31 E B(f32), there exists a <g(f32) such thatfp,(a) = f31' The restriction offp2 to lO(a) is an ordering function of B(f31)' Since f P1 : lO(g(f31» ----> B(f3d is also an ordering function, it follows by Lemma 2 that lO(a) = 10 (g(f3 1» and therefore g(f31) = a <g(f32)'


(2) The image offoe function 9 is an O-segment.

Proof Let 13 E B and (X <g(f3). We have to prove that (X E g(B). Since fp is a map of O(g(f3)) onto B(f3) and (X <g(f3) there exists 130 E B(f3) withfp«(X) = 130. The restriction ofjp to O«(X) is an ordering function of B(f3o). Sincefpo: O(g(f3o)) ~ B(f3o) is also an ordering function, so by Lemma 2 (X = g(f3o) and therefore (X E g(B).

From (1) and (2) g·is an order preserving map of B onto an O-segment A. The inverse of 9 is an order preserving map of A onto B and therefore an ordering function of B.

Theorem 13.4. Every set B £ 0 has exactly one ordering function.

Proof If B(f3) is a proper segment of B then every proper segment of B(f3) is a proper segment B(f3o) of B for some 130 < 13. It follows from Lemma 3 by transfinite induction on 13 that every proper segment B(f3) of B has an ordering function. By Lemma 3, B then has an ordering function. By Lemma 2 this is uniquely determined by B.

Definitions: A set B£ 0 is said to be closed if, for every non-empty denumerable set M,

M£B=> sup MEB.

An ordering functionf: A ~ B is said to be continuous if A is closed and, for every non-empty denumerable set U£A,

sup f( U) = f(sup U).

Theorem 13.5. The ordering function of a set B£ 0 is continuous if, and only if, B is closed.

Proof Letf:A ~ B be the ordering function of B. 1. Suppose f is continuous. For every non-empty denumerable set M£B

there is a non-empty denumerable set U £ A such that f( U) = M. Since f is continuous, sup M = f(sup U) and therefore sup ME B. Hence B is closed.

2. Suppose B is closed. Let U be a non-empty denumerable subset of A. Then f( U) is a non-empty denumerable subset of B. Since B is closed, supf( U) E B. Hence there exists (X E A such that f( (X) = sup f( U). It follows that for all e E U, fie) ~f((X) and therefore e ~(X. Hence sup U ~(X. Since A is an O-segment it follows that supUEA. Hence A is closed. For all eEU, e~supU, thereforef(e)~ f(sup U). It follows that supf(U)~f(sup U). Now sup U~(X impliesf(sup U)~ f((X)=supf(U). Hence supf(U)=f(sup U) and thereforefis continuous.

Definition. An ordering function f: A ~ B is said to be a normal function if A = 0 and f is continuous.

Theorem 13.6. The ordering function of a set B£ 0 is a normal function if, and only if, B is closed and unbounded.


Proof By Ax. II and Ax. III a subset of I[j) is bounded if, and only if, it is denumerable. It follows that if f:A ----> B is an ordering function, then the set B is unbounded if, and only if, A is unbounded, but then A = I[j). This together with Theorem 13.5 yields the assertion.

5. Addition of Ordinals

For a E I[j) let B~ be the set of all ordinals ~ a and f~ the ordering function of B~. Obviously the set B~ is closed and unbounded. So by Theorem 13.6f~ is a normal function. In this way f~(P) is defined for all a, 13 E I[j). We write a + 13 for f~(f3). This definition of addition immediately yields

(+1) a~a+f3 ( + 2) 13 < y =;. a + 13 < a + y (right strict mono tonicity) ( + 3) If a ~ 13 there is a unique ~ such that a + ~ = 13.

Since f~ is a continuous function we have

(+4) a+sup U=sup (a+ U)

for every non-empty denumerable set U c I[j) where we write a + U for {a+f3:f3E U}.

If 13 is a limit number then l[j)(f3) is a non-empty denumerable set and sup l[j)(f3) = 13· ( + 4) therefore yields in particular

( + 5) a + 13 = sup (a + l[j)(f3» for every limit number 13·

We also easily obtain

(+6) a+O=a (+7) a+f3'=(a+f3)' (+8) f3~a+f3 (by Theorem 13.3).

Since Bo = I[j),/o is the identity and hence

(+9) 0+13=13.

We also have associativity

(+10) (a+f3)+y=a+(f3+y).

Proof The ordering function for the set Bd p of all ordinals ~ a + 13 is such that fdP(y)=(a+f3)+y. Let C~.p:={a+(f3+Y):YEI[j)} and g~.p(y):=a+(f3+y). It follows from ( + 2) that the function g~. p is strictly monotone. Its image is the set C~. p. Hence g~.P is the ordering function of C~. p. It follows from ( + 1)-( + 3) that


~ E Ca.,fJ if, and only if, lY..+f3~~. Hence Ca.,fJ=Ba.+fJ· Thereforefa+fJ and ga.,fJ are ordering functions for the same set Ba.+fJ' By Theorem 13.4 it follows thatfa.+p= ga.,fJ and hence (lY..+f3)+Y=IY..+(f3+y)·

We also have left weak monotonicity

(+11) lY..~f3=lY..+y~f3+y.

Proof It follows from IY.. ~f3 by ( + 3) that there exists ~ with IY.. + ~ = 13. By ( + 8) Y ~~ +y. From (+2)and (+ 10) it follows thatlY..+Y ~IY..+(~ +y)= (IY..+ 0 +Y= 13 +y.

From ( + 6) and ( + 7) one sees that addition of finite ordinals corresponds to addition of natural numbers. If IY.., 13 E N then IY.. + 13 EN.

Remark. Addition of ordinals is not commutative, since for example we have: 0' +w=sup (0' + N)=sup N =W and w<w'=w+O' and therefore 0' +w<w+O'.

Also left strict monotonicity fails since for example 0 < 0' but 0 + w = w = 0' + w.

Definition. An ordinal IY.. is said to be an additive principal number if IY.. # 0 and ~+IY..=IY.. for all ~<IY...

Obviously 1 : =0' is the smallest additive principal number.

w is the smallest additive principal number > I.

Proof If ~<w then ~E N and ~+w=sup(~+N)=sup N=w. Hence w is an additive principal number. If 1 < IY.. < w then IY.. E Nand 1 + IY.. = IY.. + I > IY.. so IY.. is not an additive principal number. Hence w is the smallest additive principal number> 1.

If IY.. is an additive principal number then the set O(IY..) is closed under addition. That is: ~<IY.. and 11<1Y.. imply ~+I1<IY...

Proof If 11 < IY.. then ~ + 11 < ~ + IY... If IY.. is an additive principal number and ~ < IY.. then ~ +IY..=IY.., and therefore ~ +11 <IY...

Lemma 4. The set of additive principal numbers is closed and unbounded.

Proof of unboundedness. Let IY.. be an arbitrary ordinal. We have to prove that there isan additive principal number >IY... Let 130 :=IY..', f3n+ 1: =f3n+f3n' M: = {f3n: n EN} and 13= sup M. 'Then f3n<f3n+l and lY..<f3o<f3 and therefore 13#0. If ~<f3 then there exists n E N such that ~ < f3n. It follows that for all m ~ n, ~ + 13m ~ 13m + 13m = 13m + 1 <13· Hence sup (~+ M) ~f3. From ( + 8) and ( + 4) it follows that f3~ ~ + 13 = sup (~ + M) ~ 13. Hence ~ + 13 = 13 for all ~ < 13· 13 is therefore an additive principal number >IY...

Proof of closure. Let M be a non-empty denumerable set of additive principal numbers. Then sup M #0. We have to prove that sup M is an additive principal number. If ~ < sup M then there exists IY.. E M such that ~ < IY... For all 13 E M which are ~ IY.. it follows that ~ < 13 and, since 13 is an additive principal number, ~ + 13 = 13. Hence ~ + sup M = sup M for all ~ < sup M. Therefore sup M is an additive principal number.


Definition. Let 0( ~ of be the ordering function of the additive principal numbers. It follows from Lemma 4 and Theorem 13.6 that this function is a normal

function so w~ is defined for all 0( E (I) and we have: (wi) O<w~ (w2) ~<w~ => ~+w~=w~

(w3) O«P=>w~<wfJ (w4) For every additive principal number P there exists 0( such that w~ = p.

By continuity we have (w5) wsupu =sup W u

for every non-empty denumerable set UC (I). In particular we have (w6) wfJ =sup WO(P) for every limit number p.

From (w2) and (w3) we have (w7) O«P => w~ +wP =wp.

Since I is the smallest additive principal number and w is the next smallest additive principal number we have (w8) wo= I (w9) w 1 =W.

Theorem 13.7 (Cantor Normal Form). For every ordinal 0(#0 there are unique 0(1 ~ .•• ~ O(n (n~ I) such that o(=W~l + ... + w~n.

Proof of existence by transfinite induction on 0(.

• 1. Suppose 0( is an additive principal number. By (w4) there is then an 0(1

such that 0( = W~l. 2. Suppose 0( is not an additive principal number. Then there exists ~ < 0( such

that ~ + 0( # 0(. It follows that ~ # 0 and 0( < ~ + 0(. Then there exists", # 0 such that ", < 0( and 0( = ~ +",. By the induction hypothesis there are 0(1 ~ ••• ~ O(m (m ~ I) such that~ =wa1 + ... + warn and thereexist.B1 ~ ... ~ Pn(n~ I) such that", =wfl1+ ... +wfln . If we had O(i <P1 for all i = I, ... , m then it would follow from (w7) that ~ +", =", contradicting", < 0(. Hence there is a largest number k ~ m such that O(k ~ Pl' It follows that 0(1 ~ ••• ~ O(k~ P1 ~ ... ~ Pn and by (w7)

Uniqueness proof Suppose o(=W~l + '" +w~rn=wfll + ... +wPn where 0(1 ~ '" ~O(m

(m~l) and P1~"'~Pn (n~I). We prove by induction on m that m=n and O(i = Pi for all i = 1, ... , m. It follows from the properties of additive principal numbers that o«W~l", 0«wfl1 ", 0(1 <P1" and P1 <0(1'. Hence 0(1 ~P1 and P1 ~0(1 and therefore 0(1=P1' Hence either m=n=1 or m>l, n>1 and w~2+···+w~rn= wfJ2 + ... + wfJn. The assertion now follows from the induction hypothesis.

6. O(-Critical Ordinals

Definition. We define a set Cr (0() c (I) and a function <P~ for each ordinal 0( inductively as follows:


1. Cr (0) is the set of additive principal numbers. 2. 4J1l :AIl -+ Cr (a) is the ordering function ofCr (a). 3. If a =I: 0, Cr (a) is the set of common fixed points of all functions 4J~ with e < a.

That is: "E Cr (a) if, and only if, "E A~ and 4J~(,,) =" for all e < a. We call the elements of Cr (a) the a-critical ordinals. We write 4Jap instead of

4JiP) for P E All'

Corollaries. 1. a <p:;. Cr (p) ~Cr (a). 2. Every ordinal4Jap is an additive principal number. 3. 4JOp=aJI.

Lemma 5. The set Cr (a) is closed and unbounded.

Proof by transfinite induction on a. If a=O the assertion holds by Lemma 4. Now suppose a =1:0. If e <a it follows from the induction hypothesis by Theorem 13.6 that 4J~ is a normal function and therefore A~=O. .

Proof of unboundedness. Let P be an arbitrary ordinal. We have to show that there is an a-critical ordinal> p. Let Yo: = P', y,,+ 1: = sup {4Jey,,: e <a}, U: = {y,,: n E N} and y:=sup U. Then P<Yot5;,y. If e<a then 4Jey"t5;,y,,+lt5;,y so sup4JeUt5;,y. Therefore 4Jey = y for all e < a. Hence y is an a-critical ordinal > p.

Proof of closure. Let M be a non-empty denumerable subset ofCr (a). Then for all e < a and" E M we have 4Je" =". Since 4Je is continuous it follows that 4Je(sup M) = sup M for all e < a and therefore sup M'E Cr (a). Hence Cr (a) is closed.

Theorem 13.8. For every ordinal a, 4J1l is a normal function and All = O.

Proof This follows from Lemma 5 and Theorem 13.6.

Theorem 13.9. 4Ja1P1 = 4Ja2P2 holds if; and only if, one of the following holds: 1. a1<a2 and P1=4Ja2P2, 2. a1 =a2 and PI =P2' 3. a2 <a1 and 4Ja1P1 =P2'

Proof 4J(1.2P2ECr(a2)· If a1<(1.2 then 4Ja1(4J(1.2P2)=4J(1.2P2 therefore 4Ja1P1= 4Ja2(1.2 if, and only if, PI =4Ja2P2' The case of a2 <a1 is similar. For a1 =a2 the assertion is trivial.

Corollaries. 1. "E Cr «(1.') ~ 4Ja" = ", 2. If P is a limit ordinal then Cr (P) is the intersection nll<1I Cr (a) of all Cr (a)

with a<p.

Theorem 13.10. 4Ja1Pl <4Ja2P2 holds if, and only if, one of the folio wing holds: 1. a 1 < (1.2 and P1 < 4Ja2P2' 2. (1.1 =(1.2 and PI <P2' 3. (1.2 <a1 and 4J(1.1P1 <P2'

13. Theory of Ordinals of the I st and 2nd Number Classes 83

Proof ¢rx2/32 E Cr (rx2)· If rx 1 <rx2, then <Prx1(¢rx2/32)=¢rx2/32' therefore <Prx1/31 < ¢rx2/32 if, and only if, /31 <¢rx2/32. The case of rx2 <rx1 is similar. For rx 1 =rx2 the assertion follows from the fact that ¢a! is an ordering function.

Theorem 13.11. rx ~ <prxO.

Proof by transfinite induction on rx. If e < rx then ¢eO < ¢e( <prxO) = ¢rxO. By the induction hypothesis it follows that e < ¢rxO for all e < rx. Hence rx ~ ¢rxO.

Corollary. /3 E Cr (rx) ---> rx ~ /3.

Proof If /3 E Cr (rx) then there exists I'f such that /3 = ¢rxl'f. It follows that rx ~ ¢rxO ~ ¢rxl'f = /3.

Theorem 13.12. For every additive principal number'}' there exist unique rx and /3 <'}' such that'}' = ¢rx/3.

Existence proof By Theorem 13.11, '}'~¢'}'O. Since O<,}" '}' <¢'}"}'. Hence there is a least ordinal rx ~'}' such that'}' # ¢rx'}'. If rx #0 then ¢e'}' ='}' for all e < rx and therefore '}' E Cr (rx). By the hypothesis'}' E Cr (0). In any case therefore'}' E Cr (rx). Therefore there exists /3 such that'}' = ¢rx/3. Since,}, # ¢rx'}' it follows that'}' < ¢rx'}' and /3 < '}'.

Ul'Iiqueness proof If /31 <,}" /32 <'}' and ,},=¢rx1/31 =¢rx2/32 then by Theorem 13.9 rx 1 = rx2 and /31 = /32·

Definition. An ordinal rx is said to be strongly critical if rx E Cr (rx).

Theorem 13.13. rx is strongly critical if, and only if, ¢rxO = rx.

Proof If rx E Cr (rx) then there exists /3 such that rx=¢rx/3. Now from rx~¢rxO (by Theorem 13.11) it follows that /3 = 0 and hence ¢rxO = rx. Conversely: if ¢rxO = rx then rx E Cr (rx).

Theorem 13.14. The set of strongly critical ordinal numbers is closed and unbounded.

Proof of unboundedness. Let rx be an arbitrary ordinal. We have to prove that there is a strongly critical ordinal > rx. Let /30 : = rx', /3. + 1 : = ¢ /3.0, U: = {/3.: n EN} and /3: =sup U. Then rx</3o ~/3 and by Theorem 13.11 /3.~/3n+1· If e </3 then there exists n E N such that e < /3n. It follows that for all m ~ n we have e < /3m and

From this it follows that sup ¢e U ~ /3. Since ¢~ is a normal function it follows that ¢e/3 = /3 for all e < /3, and therefore /3 E Cr (/3). Hence /3 is a strongly critical ordinal >rx.


Proof of closure. Let M be a non-empty denumerable set of strongly critical ordinals. We have to prove that sup M is strongly critical. If ~<sup M there exists a e M such that ~ < a. It follows that ljJ~P = P for every P e M which is ~ a since every P is strongly critical. Hence ljJ~(sup M) = sup M for all ~ < sup M and therefore sup Me Cr (sup M). Thus sup M is strongly critical.

Following S. Feferman we denote the least strongly critical ordinal by r o.

. 7. Maximal a-Critical Ordinals

Definition. An ordinal ')I is said to be maximal a-critical if ')I e Cr (a) and ')I f/= Cr (~) for all ~ > a.

Obviously an ordinal ')I is maximal a-critical if, and only if, there exists P < ')I such that ')I = ljJap. It therefore follows from Theorem 13.12 that for every additive principal number ')I there is an ordinal a such that ')I is maximal a-critical.

We define ifJap in such a way that ifJ«, where ifJ«(p):=ifJap, is the ordering function of the maximal a-critical ordinals.

Definition. If there exists Po and n e N such that P = Po + nand ljJapo = Po let ifJap : = ljJaP'. Otherwise let ifJap : = ljJap.

Proof By definition P* = P or P* = p'. If ljJap = p then P* = p'. If P* = p' then p ~ ljJap < ljJap* and 0' ~ ljJap < ljJap*. Since ljJap* is an additive principal number it

, follows that P* = P + 0' < ljJap*. On the other hand, if P* = P then p < ljJap and therefore P* < ljJap*.

Lemma 7. ifJap is a maximal a-critical ordinal.

Proof ifJap = ljJap* where P* = P or P* = p'. By Lemma 6 P* < ljJap*. Consequently ifJap is a maximal a-critical ordinal.

Proof ifJaPi = ljJaPt where Pt =Pi or Pt = p; (i = 1, 2). Now P 1 < P2 implies Pt ~ /12· If P1=P2 then P1=P'1 and P1=PO+n where ljJapo=po and neN. But then P2=PO+n' and P~=P~· In every case therefore P1<P~. Hence ljJaP1<ljJap~ and therefore ifJaP1 <ifJaP2·

Theorem 13.15. For each additive principal number ')I there are unique a and P such that ')I=ifJap.

Existence proof By Theorem 13.12 there exist a and P1 <')I such that ')I=ljJaP1. If ifJap 1 = ljJap 1 then the assertion is satisfied with P : = P 1. Otherwise there exist


Po and n E N such that P1 = Po + nand </Japo = Po· Since P1 < </JaP1 it follows that n =F O. Therefore there is an mEN such that m' = n. Putting P : = Po + m we obtain I/Iap = </Jap' = y.

Uniqueness proof Suppose I/Ia1P1 =I/Ia2P2. By definition there exist pr such that I/IaiPi = </JaiPr (i = 1, 2). By Lemma 6 Pr < </JaiPr (i = 1, 2). Then by Theorem 13.9 </Ja1Pt=</Ja2P! implies a1 =a2' By Lemma 8 it follows that P1 =P2'

Theorem 13.16. The function 1/1 II. given by 1/1 rz<P) : = I/Iap is the ordering function of the maximal rt.-critical numbers.

Proof This follows from Lemmata 7, 8 and Theorem 13.15.

Remark. By Lemmata 7 and 8 the set of maximal a-critical ordinals is unbounded. But by Theorem 13.5 it is not closed since its ordering function 1/111. is not continuous as the following shows.

If e < </Ja'O then by Theorem 13.10 </Jae < </Ja'O and hence </Jae ¢ Cr (a') and e < </Jae. It follows that there is no e < </Ja'O with </Jae = e and hence I/Iae = </Jae by the definition of 1/111.' Setting U: = O( </Ja'O) it follows that sup I/IaU = sup </JaU. Since </JIZ is continuous and sup U = </Ja'O it follows that sup I/Ia U = </Ja'O. But I/Ia(</Jrt.'O) = </Ja(</Ja'O)' > </Ja'O and therefore I/Ia(sup U»supl/iaU. Hence 1/111. is not continuous.

THeorem 13.17. 1. If y = I/Iap then a <y if, and only if, y is not a strongly critical ordinal.

2. For all p, P < I/Iap.

Proof I/Iap = </Jap* where P* = P or P* = p'. 1.1. Suppose y is a strongly critical ordinal. By Theorem 13.13 we then have

y = </JyO. In this case I/IyO = </JyO. But then y = I/IyO = I/Iap implies y = a by Theorem 13.15.

1.2. Suppose y is not a strongly critical ordinal. Then y < </JyO by Theorem 13.11 and 13.13. By Theorem 13.10 y=</Jap*<</JyO implies a<y.

2. By Lemma 6 P* < </Jap* = I/Irt.p. It follows that 13 ~ 13* < I/Iap.

Theorem 13.18. I/Ia1p 1 < I/Ia2p 2 holds if, and only if, one of the following holds: 1. a1<a2 and 131 <I/Ia2p2' 2. rt.1=a2 and 131<132' 3. rt.2 <a1 and I/Ia1p1 ~p2'

Proof 1. Suppose a1 <a2 and 131 <I/Ia2p2' Then I/Ia2p2 is an additive principal number> 1. Therefore 131 <I/Irt.2p2 implies 13'1 <I/Ia2p2 and then I/Ia1p1 ~</Ja1p'1 < </Ja1(I/Ia2p2)' Since I/Ia2p2ECr(a2) and a1<rt.2 we have </Ja1(I/Ia2p2) = I/Ia2p2' Therefore I/Ia1p1 < I/Ia2p2'

2. Suppose a1 =a2 and 131 <132' Then by Lemma 8 I/Ia1p1 <I/Ia2p2' 3. Suppose I/Ia1p1 ~p2' By Theorem 13.17 132 <I/Ia2p2' Therefore I/Ia1p1 <

I/Ia2p2'


4. Suppose none of cases 1-3 holds. Then eitheu l =a1 and Pl = P1 so "'alPl = ",alPZ or one of cases 1-3 for ",azpz < "'alP 1 holds.

§ 14. A Notation System for the Ordinals < r 0

1. Definition of Ordinal Terms

By Theorems 13.7, 13.15 and 13.17 every ordinal -# ° which is not strongly critical can be expressed in terms of + and",. Accordingly we introduce a set T of terms denoting the ordinals < r 0 and at the same time we formally fix their lengths in the following way.

Inductive Definition of the set T of terms. I. The symbol ° is a term in the set T with length 0. 2. If al, ... ,an,pl> ... ,Pn(n~l) are terms in the set T whose lengths are

Lal, ... , Lan' LP1' ... , LPn then (a l , Pl) ... (an, Pn) is a term in the set T of length

We denote the length of a term a E T by La. This length is a natural number.

Interpretation. 1. The term ° denotes the ordinal 0. 2. If al> ... , an' P l' ... , Pn (n ~ I) are terms in the set T which denote ordinals

al , ... , al, 131, ... , Pn then the term (al> Pl) ... (an, Pn) denotes the ordinal "'alPl + ... + "'anPn'

These terms in the set T do not give a system of unique notations for ordinals. For example the terms

(0, (0, 0» and (0, 0)(0, (0, 0»

both denote the same ordinal (J) = ",0(",00) = ",00 + "'O( ",00) according to the given interpretation.

In order to obtain a system of unique notations for the ordinals < rowe single out a subset of OT of T. At the same time we inductively define the < -relation.

Inductive Definition of a set OTc T of ordinal terms and a relation < on ~T. I. The symbol ° is an ordinal term in the set ~T. 2. If al, ... ,an,Pl, ... ,Pn (n~l) are ordinal terms in the set OT and

(ai+l,Pi+l)~(a;,Pi) for all i=I, ... ,n-I then (al,Pl) ... (an,Pn) is an ordinal term in the set OT.

We call the ordinal terms (a, p) E OT principal terms. 3. If a E OT then a < 0 does not hold. 4. If P E OT then ° < P if, and only if, P -# O.

14. A Notation System for the Ordinals <r 0 87

5. If (lXI' PI)' (1X2,P2)EOT then (1X1,/31)«1X2,P2) exactly when one of the following holds:

5.1. 1X1<1X2 and Pl«1X2,P2), 5.2. IXI =1X2 and IXI <P2, 5.3. 1X2 <IXI and (lXI' Pl)~P2. 6. If Y=Yl ... YmEOT and t5=t5 1 ••• t5n EOT where Yl, ... ,Ym, t5 1, ••• ,t5n (m~l,

n ~ I, m + n > 2) are principal terms then Y < t5 exactly when one of the following holds:

6.1. m<n and Yi=t5i for all i= I, ... , m. 6.2. There existsj~min (m, n) such that Yj<t5j and Yi=t5i for all i= I, ... ,j-1.

IX=P means that IX and P are identical terms. IX~P means that IX<P or IX=P. We also write IX> P for P < IX and IX ~ P for P ~ IX.

In the sequel only the elements of the inductively given set OT are called ordinal terms.

It is clear from §13 that the. inductive definition ofthe <-relation corresponds to that in the interpretation we gave of the ordinal terms as notations for ordinals < r o. The principal terms denote the additive principal numbers < r o.

In this section we started out from the point of view of the non-constructive theory of ordinals developed in §13. Now the ordinal terms are to be understood without reference to their interpretations and simply as formal strings of symbols for which we have a constructively defined < -relation. All the theorems about orpinal terms in this section will be proved constructively using only the above inductive definition. In this way we lay a constructive foundation for the proof theoretic applications in the next chapter.

We first confirm that we are dealing with decidable properties of ordinal terms. It is obviously decidable for each string of symbols whether it is a term of the set T. By the following theorem it is also decidable whether a string of symbols is an ordinal term of the system OT.

Theorem 14.1. (1) It is decidable whether IX < P, IX = P or P < IX for aI/IX, P E OT. (2) If YET it is decidable whether Y E OT.

Proofby induction on the maximum of La., LP and Ly and within that by induction on LIX+Lp.

(I) For e,,, E OT let E(e,,,) denote: It is decidable whether e <", e =" or,,<e. We prove E(IX, P) for IX, P E ~T.

1. Suppose IX or P is O. Then E(IX, P) is trivial. 2. 1X=(1X1, 1X2), P=(Ph P2). Then by the induction hypotheses we have

E(IXI, PI), E(1X2, P2), E(1X2, P) and E(IX, P2). Now E(IX, P) follows by the inductive definition of the < -relation.

3. 1X=1X1 ••• lXm, P=Pl ... Pn where IXh ••• ,lXm, Pl, ... ,Pn (m~l, n~l, m+n>2) are principal terms. Then by the induction hypotheses E( IX;, Pi) for all i ~ min (m, n). If lXi = Pi for all i ~ min (m, n) then IX < P, IX = P or P < IX according as m < n, m = n or n<m. If there is a least indexj~min (m, n) such that IXj=FPj then IX<P or P<IX according as IXj<Pj or Pj<lXj .


(2) If y=O then Y EOT. Otherwise Y=(lXl, Pl)"'(lXn, Pn) (n~ 1). By the main induction hypothesis it is decidable whether IX; E OT and P; E OT for all i = I, ... , n. If this is the case, then we also have (IX;, Pi) E ~T. If n> 1 then by the induction hypotheses it is decidable whether (1X;+l,P;+l)~(IX;,P;) for i=I, ... ,n-1. Hence Y E OT is decidable.

In the following parts of this section small Greek letters will always denote ordinal terms in the system ~T.

2. The Ordering of the Ordinal Terms

Theorem 14.2. The ordinal terms are linearly ordered by the < -relation, that is, for all y, Yl' Y2' Y3 E OT we have:

(I) Not Y<Y (2) Yl <Y2 or Yl =Y2 or Y2 <Yl

(3) Yl <Y2 and Y2 <Y3 imply Yl <Y3'

Proof (1) follows at once by induction on Ly and (2) by induction on LYl +LY2' Using (2) we prove (3) by induction on LYl +LY2 +Lh We only give the details of the proof of (3) for the case Y; =(IX;, Pi) (i= I, 2, 3) since in the other cases the assertion follows easily from the induction hypothesis. The hypotheses Yl <Y2 and Y2 < Y3 produce one of the following seven cases:

1. IXl < 1X2' Pl <Y2 and 1X2 ~1X3' By I.H. IXl <1X2 and 1X2~1X3 yield IXl <1X3'

By I.H. Pl <Y2 and Y2 <Y3 yield Pl <h

IXl < 1X3 and Pl <Y3 yield Yl <Y3' 2. IXl < 1X2, Pl <Y2 and 1X3 < 1X2' Y2 ~P3'

By (2) either IXl <1X3 or IXl =1X3 or 1X3 <lXl'

2.1. Suppose IXl < 1X3'

By I.H. Pl <Y2 and Y2 <Y3 imply Pl <h

IXl < 1X 3 and Pl <Y3 imply Yl <Y3' 2.2. Suppose IXl = 1X3 .

By I.H. Pl <Y2 and Y2 ~P3 imply Pl <P3' IXl = 1X 3 and Pl <P3 imply Yl <Y3'

2.3. Suppose 1X3 < IX l •

By I.H. Yl <Y2 and Y2 ~P3 imply Yl <P3' 1X3 <lXl and Yl <P3 imply Yl <Y3'

3. IXl =1X2' Pl <P2 and 1X2 < 1X3' P2 <Y3' Then IXl <1X3' By I.H. Pl <P2 and P2 <Y3 imply Pl <h

IXl < 1X 3 and Pl <Y3 imply Yl <Y3' 4. IXl =1X2' Pl <P2 and 1X2 = 1X3' P2 <P3'

Then IXl =1X3 . By I.H. Pl <P2 and P2 <P3 imply Pl <P3' IXl = 1X 3 and Pl <P3 imply Yl <Y3'

5. 1X2 ~ IXl and 1X3 < 1X2' Y2 ~ P3' By I.H. 1X3 < 1X2 and 1X2 ~ IXl imply 1X3 < IX l ·

14. A Notation System for the Ordinals < r 0

By I.H. 11 <12 and 12 :!;"P3 imply 11 <P3· 1X3 <IXI and 11 <P3 imply 11 <13·

6. 1X2 <lXI' 11 :!;"P2 and 1X2 = 1X3' P2 <P3·

89

Then 1X3 <IXI. By I.H. 11 :!;"P2 and P2 <P3 imply 11 <P3· 1X3 <IXI and 11 <P3 imply 11 <13·

7. 1X2 <lXI' 11 :!;"P2 and 1X2 <1X3, P2 <13· By I.H. 11 :!;"P2 and P2 <13 imply 11 <13-

Theorem 14.3. (1) P < (IX, P) (2) IX < (IX, Pl.

Proof of (1) by induction on Lp. 1. P=O. Then the assertion is trivial. 2. P=(Pl' P2)· ByI.H. wethenhaveP2 <pandP2 «IX, P2). Now P2 <pimplies

(IX, P2)«IX, P)· Hence fJ2 «IX, Pl. 2.1. Suppose PI <IX. Since P2 «IX, P) we then have P«IX, P). 2.2. Suppose PI = IX. Since P 2 < P we then have P < (IX, P). 2.3. SupposelX<Pl. Since p:!;,.pwe have P«IX,Pl. 3. P=Pl ... Pn (n~2) where PI' ... , Pn are principal terms. By I.H. PI «IX, PI).

Since PI < P it follows that PI < (IX, P). Hence P < (IX, Pl.

Proof of (2) by induction on LtX. I. IX = O. Then the assertion is trivial.

• 2. 1X=(1X1, 1X2). By I.H. we then have 1X1 <IX and 1X2 < (1X2' Pl. By (1) 1X2 <IX and P < (IX, Pl. Hence (1X2' Pl < (IX, Pl. Therefore 1X2 < (IX, Pl. Now 1X1 < IX implies IX < (IX, Pl.

3. 1X=1X1 ... lXn (n~2) where 1X1, ... ,lXn are principal terms. By I.H. 1X1 «1X1,P). By (1) P«IX, Pl. Since 1X1 <IX it follows that (lXI' P)«IX, Pl. Hence 1X1 «IX, Pl. It follows that IX < (IX, Pl.

Remark. We require ordinal terms in the succeeding chapters in order to carry out induction over OT with respect to the < -relation. That is to say: we use the principle of proof by transfinite induction over the segment of the ordinals < r 0

represented by ~T. However, the well-ordering of OT, on which this transfinite induction is based, cannot be proved in as elementary a way as Theorem 14.2. Later we shall establish transfinite induction for OT, using stronger methods step by step.

3. Addition of Ordinal Terms

Definition of IX + P E OT and IX' E OT for IX, P E ~T.

1. IX+O:=lXandO+p:=p. 2. If IX = 1X1 ... lXm and P = Pl·· ·Pn with principal terms 1X1,···, IXm' PI'···' Pn

(m~l,n~l)set

2.1. IX+P :=P, ifIXl <PI' 2.2. IX + P : = 1X 1 .. . lXkP 1 .. . Pn, if k is the largest index :!;,.m such that PI :!;,.lXk. 3. IX': = IX + (0, 0).


The following properties are easily established for this addition of ordinal terms.

1X~IX+p and p~lX+p,

P<y => 1X+/3<IX+Y, lX~p => lX+y~/3+y,

IX + (13 + y) = ( IX + 13) + y. If IX = 1X1" .lXn (n ~ 2) where 1X 1, "., IXn are principal terms IX = 1X1 + ". + IXn'

IX' is the successor of IX since we obviously have

IX < IX' IX < /3 => IX' ~ /3

lX<p' => 1X~/3.

The principal terms have the properties of additive principal numbers since the definition of addition yields

~ < (IX, 13) => ~ + (IX, 13) = (IX, 13) ~ «IX, /3),,,, «IX, /3) => ~ +", < (IX, 13)·

Definition of -IX + 13 for IX ~ p. I. -IX+IX:=O 2. -0+13 :=/3 3. If 1X=1X1."lXm<p=Pl."Pn with principal terms IX b ,,·, IXm, 131,"" /3n (m~ 1,

n ~ 1) we consider the following two cases corresponding to the definition of the < -relation:

3.1. m<n and IX j=/3j for all i= 1, "., m. In this case set -1X+p :=/3m+l".Pn' 3.2. There existsj~min (m, n) such that IXj<pjand IXj=pj for all i= 1, ".,j-l.

In this case set -IX + 13 : = /3j" ·Pn.

In every case - IX + 13 is again an ordinal term. The definition yields

lX~p => IX+( -1X+p)=p.

4. Ordinal Terms 4>1X/3

We call the ordinal term 0 and the ordinal terms 1X1".lXn (n~ 1) such that IXj=(O, 0) (i = 1, ... , n) finite ordinal terms.

Definition. w: =(0, (0,0».

Obviously IX < w holds if, and only if, IX is a finite ordinal term.

Definition of 4>lXp for ordinal terms IX, 13: I. If /3= (pb pz) where lX<pl, then set 4>lXp : = p. 2. If 13=(131, /3z)+y' where lX<pl and y<w, then set 4>1X/3 : = (IX, (/31, Pz)+Y). 3. In every other case set 4>1X/3 : = (IX, 13).

14. A Notation System for the Ordinals <To

Corollary. 4Ja.p is a principal term.

Theorem 14.4. (l) P ~ 4Ja.p (2) a. < 4Ja.p.

91

Proof of (1). If 4Ja.p = P then the assertion holds. If 4Jrxp = (rx, p) then P < 4Jrxp by Theorem 14.3(1). Otherwise we have P=(PI,P2)+Y' where rx<PI and y<w and 4JrxP=(rx,(PI,P2)+Y)' In this case (0,0)«PI,P2)+Y' By Theorem 14.3(1) we

, obtain (P b P 2) + Y < 4Jrxp and (0,0) < 4Jrxp. Hence P < 4Jrxp.

Proofof(2). If P=(PI' P2) where rx<PI then 4Jrxp=p. By Theorem 14.3(2) we then have PI <po It follows that rx<4Ja.p. Otherwise we have 4Jrxp=(rx, Po). Then by Theorem 14.3(2) again we have rx < 4Jrxp.

Lemma 1. P < Y => 4Jrxp < 4Jrxy.

Proof Following the definition we have the following three cases to consider for 4Jrxp and 4Jrxy.

1. 4Jrxp = p. Then P < Y implies 4Jrxp < 4Jrxy since Y ~'4Jrxy by Theorem 14.4( 1). 2. 4Jrxp = (rx, Po) where P=Po or P=P'o and 4JrxY=Y=(YI, Y2) where rx<YI'

Now P<y implies Po<y. Using rx<YI it follows that 4Jrxp=(rx, Po) <y=4Jrxy. 3. 4Ja.p = (rx, Po) where P=Po of P=P'o and 4Jrxy=(rx, Yo) where Y=Yo ory=Y'o·

N.ow P < Y implies P ~ Yo. If P = Yo then Y = Y'o. By the definition of 4Jrxy we then have P=YO=(Yb Y2)+<> where rx<YI and <><w. Since we have 4Jrxp=(rx, Po) it follows that <>#0. Then P=P'o. In any case we therefore have Po<Yo. It follows that 4Jrxp = (rx, Po) < (rx, Yo) = 4Jrxy.

Theorem 14.5. 4Jrx i PI =4Jrx2P2 holds if, and only if, one of the following holds. 1. rx l <rx2 and PI =4Jrx2Pb 2. rx l =rx2 and PI =Pb 3. rx2 <rx l and 4Jrx i PI =P2'

Theorem 14.6. 4Ja. IPI <4Jrx2P2 holds if, and only if, one of the following holds: 1. rx l <rx2 and PI <4Jrx2Pb 2. rx l =rx2 and PI <P2, 3. rx2<rxl and 4Jrx i PI <P2'

Joint Proof for Theorems 14.5 and 14.6. By definition 4Jrx2P2 is an ordinal term (YI, Y2) with rx2 ~YI' If rx l <rx2 then rx l <YI and then again by the definition 4Jrx l (4Jrx 2P2)=4Jrx2P2' The assertions of Theorems 14.5 and 14.6 for rx l <rx2 now follow by Lemma I. The case where rx2 < rx l is similar. If rx 1 = rx2 then the assertions follow immediately from Lemma 1. These are all the cases since rx 1 <rx2 or rx 1 =rx2 or rx2 <rx 1 •

Theorem 14.7. For each principal term Y there exist unique rx and P < Y such that y=4Jrxp. Moreover, Lrx<Ly and LP<Ly.


Existence Proof We have y=(tx, Po) where Ltx<Ly. 1. If Po = (P b P2>+ (j where tx < Pl and (j < w then y = c/>txP'o· Then Po ~

c/>txpo < c/>txP'o and (0,0) < Po < c/>txP'o· It follows that P'o < c/>txP'o· Therefore LPo # 0 and LP'o =LPo + I <2· Lpo + I ~ Ly.

2. Otherwise y = c/>txPo and by Theorem 14.3 Po < c/>txpo. Then we also have LPo<Ly.

Uniqueness Proof y=c/>txlPl =c/>tx2P2 where Pl <y and P2 <y implies txl =tx2 and Pl =P2 by Theorem 14.5.

Remark. A comparison of Theorems 13.9, 13.10 and 13.12 with Theorems 14.5-14.7 shows that the ordinal terms c/>txP have the properties corresponding to those of the ordinals c/>txP defined in § 13.

Theorem 14.8. For each ordinal term y #0 there are unique ordinal terms txl,···,txn, Pl,···,Pn (n~l) where Pi<c/>txiPi (i=I, ... ,n), c/>txlPl~···~c/>tx~Pn and y = c/>txlPl + ... + c/>txnPn·

Proof We have y=Yl ... yn (n~I) with principal terms Yl~···~yn' By Theorem 14.7 there are unique txi and Pi < Yi such that Yi = c/>txi Pi' Hence the assertion is established.

5. Ordinal Terms w~, w~· n and w~· P

Definition. w~: = c/>Otx.

Corollaries. w~ is a principal term, W O = (0, 0) and w(o, 0) = W.

Lemma 2. tx < P => w~ < oJi .

Proof This follows from Lemma I.

Lemma 3. For each principal term P there is a unique ordinal term tx such that P = w~.

Proof By Theorem I4.7we have P=c/>P1P2' If Pl =0 then the assertion holds with tx: = P2' Otherwise by Theorem 14.5 it holds for tx: = p. The uniqueness of tx follows from Lemma 2.

Theorem 14.9. For each ordinal term tx#O there are unique txl ~ ... ~txn (n ~ I) such that tx=W~1 + ... +w~n.

Proof This follows from Lemma 3 and the properties of addition of ordinal terms.

Inductive Definition of w~· n E OT for tx E OT and n EN.

1. w~·O :=0 2. w~· (n + I): =w~·n +w~.

14. A Notation System for the Ordinals <To 93

Definition of v(oc) E OT and k(oc) EN for OC'E OT. 1. If oc=o then set v(oc) :=0 and k(oc): =0. 2. If oc #0 then by Theorem 14.9 there are unique OC 1 ;;:: ... ;;::ocn (n;;:: I) where

oc=afl! + ... + w~n. Then set v(oc) : = OC 1 and k(oc) to be thelargest index ~ n such that OCk(~) = oc 1 •

Theorem 14.10. (I) wV(~)·k(oc)~oc<wv(~)·(k(oc) + I)

(2) If oc <wP and P# 0 then v(oc) < p andfurther wV(~)' (k(oc) + l) <wP.

Proof Immediate from the definitions.

Definition of w~· P E OT for oc, j3 E OT. 1. w"·O:=O 2. If P # 0 then by Theorem 14.9 there are unique P 1 ;;:: ... ;;:: Pn (n;;:: I) such that

P=wP1 + ... +wPn. Then set w~· p: =WdP1 + ... +wdPn.

Corollaries: w~·(O, O)=w" P<y = w~·P<w"·y w~· P + w~· y = w~· (P + y) w~· (wp ·y)=w"+P .y.

;rheorem 14.11. For each oc and P there are unique ~ and11 <w~ such that P= w~· ~ +11.

Existence Proof I. Supposep<w~. Then theassertionholdswith~: =Oand11: =p. 2. Suppose wl1.~ p. Then there are P1;;:: ... ;;:: Pn(n;;:: 1) such that p=wP 1 + ... +wP" and OC~P1' Hence there is a largest index m~n such that oc~Pm' We set ~:= W-~+Pl+···+W-dP~ and 1'/:=0 if m=n and I'/:=wP~+I+···+wPn if m<n. In any case we then have P = w"· ~ + 1'/ and 1'/ < w".

Uniqueness Proof Suppose P=w~· ~1 +'11 =w~' ~2 +1'/2 where 111 <w~ and 1'/2 <w~. Then

It follows that ~1 <~~ and therefore ~1 ~~2' Similarly ~2~~1' It follows that ~1 =~2' Therefore 111 =112'

Lemma 4. Ify<w"· P and oc#O then y' <w"· p.

Proof y<w~·p implies y' ~w~·p. Since oc#O, w~·p is not of the form y'=y+wo. It follows that y' < w~· p.

Theorem 14.12. Ify < w~· pandoc1 < oc then there exists P1 such thaty < W~l. P1 <w~· p.

Proof By Theorem 14.11 we have y = W~l . ~ + 11 where 1'/ < W~l. Therefore


Ofl·C;~y<W~·P=W~l·(W-~lH.p). It follows that C;<W-~lH.p. Now a l <a implies 0< -al +a. By Lemma 4 it follows that C;' <W-~l +~. p. Putting Pl : =C;' we obtain y<W~l'C;+W~l=W~l'Pl <wal·(w-al+a·p)=wa.p.

6. Ordinal Terms n . p, 2~ and 2~· P

Since w(O,O)=w, it follows by Theorem 14.11 that for each a there are unique C; and 1'/ < w such that

For I'/<w, there is a unique nE N such that I'/=wo·n. We define q(a) :=C; and r(a): =n. In this way, we have q(a) E OT and r(a) EN for each a E OT such that

a=w·q(a) +wo ·r(a)

holds.

Definition of n· P for n EN:

1.0·P:=0 2. n· P :=w·q(P)+wo ·(n·r(p» for n#O.

Corollaries. I· p= P n·(p +y)=n· P +n·y n#O, P<y ~ n·p<n·y.

Definitions. 2~: = wq(~)· 2r(~), 2~· P : = wq(~)· (2r(~). Pl.

Corollaries. 2~·(0, 0)=2~

P<y ~ 2~·P<2~·y 2~ . (P + y) = 2~ . P + 2~ . Y 2~· (2 fJ . y) = 2~+fJ. y.

Theorem 14.13. If y<2~·p and a l +w~a, then there exists Pl such that y<2~1'Pl <2~·p.

Proof a l +w~a implies q(ad<q(a). Therefore, by Theorem 14.12 there exists Pl such that

It follows that

14. A Notation System for the Ordinals < r 0 95

Since -q(OCl) +q(OC) >0 and 2r(~,)<w, we have also

It follows that

7. Ordinal Terms B~, wn(oc) and 'n Definition. B~: = cfJ(O, O)oc.

These ordinal terms B~ are called B-terms. They are precisely the ordinal terms ~ such that w~ = e.

It follows from Lemma 1 that

We obviously also have

e < B~ ~ e' < B~.

Inductive Definition of wn(oc) E OT for n EN and oc E OT. 1. wo(oc): = oc 2. wn+l(oc) :=w"'n(~).

Theorem 14.14. OC<Bp ~ Wn(OC) <Bp.

Proof by induction on n. The assertion is trivial for n = O. Suppose n = m + 1. By I.H. we then have Wm(OC) <Bp. It follows that wn(oc)=w"'m(~)<W·P=Bp.

Theorem 4.15. For each OC<Bp there exists nE N and either y=O or Y=B' ~ (where 1'/<P) such that oc<wn(Y)·

Proofby induction on Loc. 1. oc=O. Then oc<wo =Wl(O). 2. Suppose oc is a principal term. Then by Theorem 14.7 there exist e and 1'/ < oc

such that oc = cfJe1'/ and L1'/ < Loc. 2.1. Suppose e = O. Since 1'/ < Bp and L1'/ < Loc by I.H. there exist n E N and a Y

of the required kind such that 1'/<wn(Y). It follows that oc=w~<w"'n(Y) =wn+l(y). 2.2. Suppose e = (0, 0). Then oc = B~ < Bp and therefore 1'/ < p. It follows that

oc<e~=wo(B~). 2.3. Suppose (0,0) < e. Then oc = B~ < Bp and therefore oc < p. It follows that

oc < B~ = Wo(B~). 3. Suppose oc=ocl ..• OCm (m~2) where OCl' .•. , OCm are principal terms. Then


~I < GfJ and L~I < L~. By I.H. there exist therefore n E N and a Y of the required kind such that ~1<Wn(Y)~Wn+I(Y). Since wn+I(Y) is a principal term it follows that ~<Wn+I(Y).

Inductive Definition of (n E OT for n EN. 1. (0: = cpOO, 2. (n+ I : = CP(nO, By Theorem 14.4(2Hn«n+ I.

Theorem 14.16. For every ordinal term ~ there exists n EN such that ~«n.

Proof by induction on L~. 1. ~=O. Then ~«o. 2. Suppose ~ is a principal term. Then by Theorem 14.7 there exist ~I' ~2

such that ~ = CP~I ~2 and L~i < L~ (i = 1,2). Then by I.H. there exist ni E N suck that ~i«ni (i= 1, 2). Putting n: = max (n l , n2 ) it follows that ~I «n and ~2 «n«n+ I.

It follows by Theorem 14.6 that ~ = CP~I ~2 < CP(nO = (n + I. 3. Suppose ~=~I ... ~m (m~2) where ~I' ... '~m are principal terms. Then

L~I <L~. Therefore by I.H. there exists nE N such that ~I «~n0 Since (n is a principal term it follows that ~«n.

8. A Mapping onto the Natural Numbers

Definitions for Natural Numbers. Ifm, n EN set n(m, n) : =t<m +n)(m +n + 1) +m, it is easy to see that n: N x N ---+ N is a bijection of the set of ordered pairs (m, n) of natural numbers onto the set N of natural numbers. Let n l and n2 be the inverses of n such that nl(n(m, n»=m, n2(n(m, n» = nand n(nl (n), n2(n» = n for all m, n E N. Just like n the functions n I and n2 are calculable arithmetic functions.

Let Po: =2. If n>O let Pn be the n-th odd prime number, thus PI = 3, P2 = 5, P3 = 7, P4 = 11, etc.

Lemma 5. If kEN then n;(k)+ 1 <pdi= 1, 2).

Proof It is easy to see that n;(k) ~ k and k + 1 <Pk. Hence the assertion follows.

Extension of OT to OT. We adjoin a symbol r 0 to the ordinal terms ofthe system OT and set

If~, P E OT then ~ < P if, and only if, either ~, P are from OT and ~ < P or ~ E OT and P = r o. Obviously OT is ordered by this < -relation.

Let N + be the set of positive natural numbers.

14. A Notation System for the Ordinals < r 0

Inductive Definition of Nr(lX) EN + for IX E OT (by induction on LIX). 1. Nr (0): = I 2. If 1X=(1X 1, Pl)"'(lXn , Pn) E OT (n~ I) let

n

Nr (IX) : = Il P1t(Nr(~;)- I, Nr(p;)- I)' i= 1

We also set 3. Nr(ro) :=0. Then Nr is a map of OT into the set N of natural numbers.

Inductive Definition of t(m) E OT for mEN + (by induction on m). 1. t(l) :=0 2. Ift(1t1(k)+ I)=IX and t(1t z(k) + I)=P, then let t(pd :=(IX, P).

n

97

3. If m = Il Pk; (n ~ 2), t(Pk;) = ')'i for all i = I, ... , nand f is a permutation of i= 1

{l, ... , n} such that')' f(1) ~ ... ~')' fin) then let t(m) : =')' f(1) ... ,), fin)'

By Lemma 5 this is indeed an inductive definition. For every prime number Pkt(Pk) is a principal term and for every natural number m~ I t(m) is an ordinal term of the system ~T. We also set

4. t(O) :=ro. Then t is a map of N into ~T.

Theorem 14.17 Nr is a bijection ofOT onto N with inverse t: N ~ ~T. That is: the following hold:

(1) t(Nr) (IX» = IX for all IX E OT (2) Nr(t(m»=m forallmEN.

Proof (I) and (2) are immediate from the definition. This yields the assertion.

The < -relation of OT induces an ordering relation -< of the natural numbers under the bijection as follows:

m-<n<:;>t(m)<t(n) for m,nEN, Nr(lX) -< Nr(p) <:;> IX < P for IX, P E OT.

Theorem 14.18. -< is a decidable arithmetic relation, that is, for any two natural numbers m, n it is decidable whether m -< n, m = nor n -< m.

Proof Given natural numbers m, n the terms t(m) , t(n) E OT are effectively calculable. We can then decide whether t(m) <t(n), t(m)=t(n) or t(n) < t(m).

Chapter VI

Functionals of Finite Type

In this chapter we treat the theory of functionals of finite type in which, following K GOdel [2] pure number theory can be interpreted (§19). The functionals concerned are introduced in §15 and they are shown to be calculable in §16 following W. Howard by transfinite induction up to 8o, in fact, by using the ordinal terms of §14. In §17 we develop a formal system on the basis of the positive implicational calculus (with equality) as a logical setting for the theory of functiorrals of finite type.

§15. The System of Terms of Finite Type

1. Types

We now introduce function types in contrast to the predicate types of Chapter IV.

Inductive Definition of types: 1. The symbol 0 is a type. (It denotes the type of the natural numbers.) 2. If (J and 't are types then «(J)'t is also a type. « (J)'t denotes the type of certain

maps (J ~ 't from the set of type (J into the set of type 't.)

For brevity we write (J't for «(J)'t, if the type (J is written as a single symbol. According to the above definition every type different from 0 is of the form

't 1 ••• 'tno (n~ I).

2. Terms

We take as primitive terms: 1. Denumerably infinitely many variables of each type. 2. The arithmetic primitive term 0 (for the natural number zero) and!/' (for

the successor function). 3. Combinators Kut and SPUt for all types p, (J, 'to (Here we use the combinators

K and S which were introduced by H. B. Curry in a type-free manner but now making distinctions of type.)

4. An iterator Jt for each type 't.

15. The System of Terms of Finite Type 99

Inductive Definition of the terms and their types: 1. Every variable of type 1: is a term of type 1:.

2. The arithmetic primitive term 0 is a term of type o. 3. The arithmetic primitive term!/ is a term of type 00.

4. The combinator Ka, is a term of type 1:CT1:. 5. The combinator Spa, is a term of type (pCT1:)(pCT)p1:. 6. The iterator J, is a term of type o( 't't)'t't. 7. If aa, and ba are terms of types CT1: and CT then aa,( ba) is a term of type 1:.

For brevity we write aa'ba for aa,( ba) if the term ba is not written as a composite term.

We call terms which contain no variablefunctionals.

Syntactic symbols: x', y', z' for variables of type 1:,

a', b', c', d', e' for terms of type 1:.

We shall also use these syntactic symbols with subscripts. The superscripts always give the type.

According to the above definition of term every term which is not primitive is of the form

where aa, .. . an, is a primitive term . . Inductive Definition of the numerals N" (n EN):

1. No :=0 2. N,,+l :=!/N". The numerals are special terms of type o.

3. Interpretation

The term ift ba denotes the image of ba under the map if': CT -+ 1:. This image is of type 1:.

The successor functional!/ maps each term aO of type 0 onto its successor !/ aO•

Thus !/ denotes a map 0 -+ 0 of type 00.

Ka.a' denotes the constant function CT -+ 1: which maps every term ba of type CT to a'. Thus Ka.a'ba =a'. This mapping Ka.at is of type CT1:. The combinator Ka, denotes the function 1: -+ CT1: which maps (J,' to the constant function K.rtrx.. Consequently Ka, is a functional of type 1:CT1:.

Spatapa'bPa denotes the function p -+ 1: which maps the term cP to the term apatcP(bPacP) of type 1:. Thus we have

Consequently Spatapa'bPa is oftype p1:, Spatapat is of type (PCT)p1: and Spa, is of type (PCT1: )(PCT) p1: •

100 VI. Functionals of Finite Type

The iterator Jt only has a meaning when together with a numeral N n• Then the functional J.Nn maps a function att: t -+ t onto its n-fold iteration. That is, the following holds:

where att occurs n times on the right-hand side of the equation. In particular we have

and

Consequently JtNnOtt is of type tt, J.Nn of type (tt)tt and Jt of type O(tt)t1;.

The interpretation we have just given serves only to clarify the theory we are dealing with. In the sequel we shall not refer to this interpretation and the meaning of the terms will be fixed purely formally by an appropriate reduction procedure.

4. Reduction Procedure for Terms

Definition. A term is said to be open if it is of one of the following three forms: l. Ka.atbadi' ... d;,.m 2. S patapat bpa cP di' ... d:,.m 3. JtNnattbtdP ... d:,.m

where in each case t=t1 ... t m+1 (m~O). The term is then oftype t m+ 1.

Inductive Definition of the reducibility of a term. I. Every open term is reducible. 2. If aat or ba is reducible then aatba is also reducible.

Inductive Definition of at [>1 bt ("at reduces to bt in one step"). The following hold for t = t 1'" tm+ 1 (m ~O): l. Ka.atbadi' ... d:,.m[>1 atdi' ... d:,.m 2. SpatapatbpacPdi' ... d:,.m [>1 apatcP(bPacP)di' .. . d;,.m

3. JtOattbtdi' ... d;,.m[>1 btdi' ... d:,.m . 4. JtNn+1attbtdi' ... d:,.m[>1 att(JtNnattbt)di' ... d;,.m 5. ba [>1 Ca => aatbadi' ... d;,.m [>1 aatcadi' ... d:"m,

provided aatbadi' ... d;,.m is not open and aat is irreducible. These reduction steps correspond to the interpretation of the terms we gave

above. Namely, at [>1 bt says that bt has the same meaning as at in the given interpretation.

Cases 1.-4. yield reductions of open terms. Case 5. gives a reduction of a reducible term which is not open.


Corollary. For each at there exists bt such that at l> 1 bt if, and only if, at is reducible. In this case bt is uniquely determined by at.

Inductive Definition of at l>" ht ("at reduces to bt in n steps") for n EN: 1. at l> Oat 2. at l>1 b\ bt l>m ct = at l>m+ 1 ct (m ~ 1).

at l>bt means that there exists n EN such that at l>" bt.

Corollaries. (1) If at l>" bt and, =, 1'" 'm+ 1 (m ~ 1) then we also have

(2) at l>bt and at l>ct implies btl>ct or ctl>bt .

Proof (1) is immediate from the defining rules for reduction. (2) holds because at most one (one step) reduction is possible for any given term.

5. Characterization of Numerals

Theorem 15.1. The numerals are the only irreducible functionals of type o.

Proof Let aO be an irreducible functional of type o. We prove by induction on the length of aO that aO is a numeral. It follows from the rules given above for types when using combinators that every term of type 0 which begins with a combinator is an open term. Hence the irreducible term aO cannot begin with a combinator. So we have at most the following three cases to consider for the term aO (which contains no variable):

1. aOisO 2. aO is !/bO 3. aO is JtbOcttdte~I".e~"where '='1.","0 (n~O). In cases 2 and 3 bO is an irreducible functional of type o. Since bO is shorter than

aO then by I.H. bO is, in these cases, a numeral. But then in case 3 aO is an open, and therefore reducible, term. So this case is excluded. In the remaining first two casesaO is a numeral.

6. Substitution Properties

We denote by c" (:: ) the term which results from c" when every occurrence of the

variable xt in c" is replaced by the term at.

Lemma 1. If c" is an open term and c"l>l d" then c"(::)l>1 d"(::}


Proof This follows at once from the rules for reduction of open terms.

Lemma 2. If xT occurs in ca and aT is reducible then ca(::) is reducible.

Proof In this case the reducible term aT occurs in ca(::) and hence the assertion

follows by induction on the length of ca.

Lemma 3. If ca is a term which is not open and does not begin with x" and if aT is

reducible then the term ca( ::) is not open.

Proof Suppose ca is a term c~cfl ... c~n where P=Pl ... Pn (n~O) and where c~ is a

basic term different from xT. Then the term ca(::) also begins with ~.

1. Suppose c~ is a variable or an arithmetic basic term. Then trivially ca(::)

is not open.

2. Suppose c~ is a combinator. Then ca(::) is not open since ca is not open.

3. Suppose c~ is an iterator.

3.1. Suppose n~2. Then trivially ca(::) is not open.

3.2. Suppose n > 2. Then PI = 0 and c~ is not a numeral since ca is not open.

If xT occurs in c~ then c~ (::) is reducible by Lemma 2. In any case c~ (::) is not a

numeral. Therefore ca(::) is not open.

Theorem 15.2. If aT 1:>1 bT and ca(::) I:>m ea where ea is irreducible then ca(~:) I:> ea.

Proof Without loss of generality we may assume that xT does not occur in aT, bT. For we can choose a variable yT which does not occur in a" b" ca. Then the terms

ca(::} cae:) are the same as the terms ca(~:)(;:} caG:)G:) so we can replace

ca and xT in the hypotheses and statement of Theorem 15.2 by ca(~:) andyT where

y' does not occur in a" bT. Now we assume that xT does not occur in aT, bT and prove Theorem 15.2 by a

multiple nested induction, namely, by induction I. on the length of the term a" 2.onm, 3. on the length of the term ca.


For brevity, in what follows we write

for any term dP•

1. Suppose ca is an open term. Then we have a term da such that ca 1>1 da. It follows by Lemma 1 that ca 1>1 da, and ca 1>1 aa. It follows from ca 1>1 aa and the hypothesis cal>m ea because of the irreducibility of ea that m > 0 and aa I>m-l ea holds. By the second LH. it follows that aal>ea. Hence cal>ea also holds.

2. Suppose ca is a term which is not open and does not begin with xt. If xt does not occur in ca then the assertion is trivial. Now assume that xt does

occur in ~. Then by Lemma 2 ca is reducible and Lemma 3 ca is not open, while at is reducible by hypothesis. Therefore ca must be of the form

where c: is irreducible and ci' is reducible. It follows by Lemma 2 that xt does not occur in c:. It now follows from the hypothesis cal>m ea that there is an irreducible termei' such that ci'l>m, ei' where 1 ~ml ~m. Sincem1 ~m and ci' is shorter than ca it follows by the second or third I.H. that ci'l>ei'. Hence

and

Since m-m1 <m it follows by the second I.H. that

But then cal>ea also holds. 3. Suppose ca is a term beginning with xt. Then ca is of the form

3.1. Suppose at is open or ca is not open. Then it follows from the hypotheses of the theorem that

By the second LH. we have

This is just the assertion ca I> ea.


3.2. Suppose at is not open and c" is open. Then it follows from at 1>1 b t that

at is a term a~ai' ... a~r

and

where p = t 1'" tr t (r ~ 1) and where a~ is irreducible and aI' I> 1 b~' holds. Let yt' be a variable which does not occur in at, bt , c".

Let dr be the term a~y"a~2 ... a~rc!' .. . c:" so that (lr 1 is the term C". Since c" ( at') yt'

is open and at, a~' are reducible it follows from Lemma 3 that di is also open. Hence we have a term di such that di I> 1 di. It follows by Lemma 1 'that

c" 1>1 J" 1 I>m-1 e" ( at') 2 yt'

and

C"1>1 J" 1 . ( bt')

2 y"

It follows from the second I.H. that

J,,(a i') I>e". 2 yt'

Since ai' 1>1 bi' holds and ai' is shorter than at it follows by the first I.H. that

J,,(bi') I>e". 2 yt'

This yields the assertion C"I> e".

7. The Normal Form of a Term

Definition. bt is said to be a normal form of a term at if at I>b t and bt is irreducible.

Corollary. Every term has at most one normal form.

Proof at I>b t and at I>ct imply bt I>ct or ct I>bt • If bt and ct are irreducible then it follows that bt and ct are identical.

In the next section we shall prove that every term has a normal form.

16. Orders of Terrns 105

Remark. The existence of the normal form can be proved somewhat more simply by using a strong metamathematical predicate (see J. R. Shoenfield: Mathematical Logic 1967, pp. 225,227). W. Howard's proof, which is given in §16, requires substantially more machinery but yields more information, since by its reference to the ordinals < Eo it gives a connection with the consistency proofs which we shall carry out in the later chapters using transfinite induction.

§16. Orders of Terms

Following a procedure of W. Howard we define the order of a term to be an ordinal < Eo which is decreased when the reduction process for terms is applied. It then follows by transfinite induction up to Eo that every term has a normal form.

We represent the ordinals by the ordinal terms we defined in §14 but in this section we restrict ourselves just to ordinal terms < Eo .

By Theorem 14.9 for every ordinal term IX#O there are uniquely determined ordinal terms IXI ~ ..• ~ IXn (n ~ 1) such that

lX=of ' + ... +o/'n.

It .follows that all IX; < IX if IX < Eo.

1. Natural Sum and Natural Product of Ordinal Terms

In order to assign ordinal terms to terms we use the natural sum IX # P and the natural product 21Z x P of the ordinal terms IX, p and we first define these.

Definition of IX # p. 1. IX # 0 : = IX and 0 # P : = p 2. If IX = W lZ1 + ... +wlZm and P=wlZm + 1 + ... +WlZn (I ~m<n) where IXI ~ ... ~lXm

and IXm + I ~ ... ~ IXn and f is a permutation of {1, ... , n} such that IX f(l) ~ ••• ~ IX fin)

then

IX# p: =WlZf (1) + ... +WlZf(n).

Corollaries.

(# 1) IX#P=P#IX (#2) 1X#(P#y) =(IX# p) #y (#3) P<y => IX#P<IX#y.

Definition of 21Z x p. By Theorem 14.11 for each IX there are unique lXo and m < w such that IX =


w·ocn+m. We regard m as a natural number so that e·2m is defined as the 2m-fold sum e + ... + e for each ordinal e.

1. 2"x 0:=0 2. If P=w(l'+···+wPn (n~l) where P1~···~Pm and oc=w·ocn+m where

m<w then set

We immediately obtain from this definition:

(xl) P~2"xP ( x 2) P < y =.> 2" x P < 2" x Y (x3) 2"x(P=II=y)=(2"xP>=II=(2"xy) (x4) 21 xP=P=II=p.

We also have

(x 5) oc<p, yl=O =.> 2" x y<2(1 x y.

Proof If oc=w·oco+m1 <P=w·po+m2 where m1 <w, m2<w then either oco<Po or oc =P m <m If oc <P then oc ..I1.y·<P ..I1.y. and w"o#Yi·2m'<w(lo#Yi Hence o 0' 1 2- 0 0 o1t' I o1t' I •

2" x y < 2(1 x y. If OCo = Po, m 1 < m2 then the assertion follows from 2m, < 2m2.

(x 6) 2" X (2(1 X y)=2,,#(1 X y.

Proof If oc=w·oco+m1 and P=w·po+m2 where m1 <w, m2<w then oc=ll=P= w·(oco=ll=po)+m1 +m2. The assertion now follows from the~efinition of natural product.

( X 7) (2" X p> =11= (2" x P) = 211# 1 X p.

Proof By (x3), (x4) and (x6), (2"xP)=II=(2"xP)=2"x(P=II=P)=2"x(21xP)= 2"#1 xp.

2. Assigning Ordinal Terms to Terms

Inductive Definition of the degree g-r of a type -r: 1. go:=O 2. g(u-r) :=max(gu+ l,g.). The degree of a type is a natural number.

Inductive Definition of an ordinal term [a']i for each term a' and natural number i. 1. If a' is a variable or an arithmetic basic term 0 or [/ then set

16. Orders of Terms

2. If a' is a combinator of type or, then set

[ t] .. ={I ifi~g-r. a.. O.f

1 i>g-r.

3. If Jp is an iterator of type -r=o(pp)pp theng-r=gp+2. Then set

{I ifi~gp+ I,

[Jp];:= w ifi=gp+2, o ifi>gp+2.

{I ifi~gp+ I,

[JpNn ];:= n if i=gp+2, o if i>gp+2.

4. If the term autbu is not of the form JpNn then set

107

This last rule is inductive since [aUtbU]gu+ 1 = [aUt ]gu+1 and therefore [aatba]; is determined for i ~g(1 by induction on g(1 - i.

Corollaries. 1. For every term a' there exists kE N such that [at];=Ofor all i>k. 2. If at does not contain an iterator then [at]; < w for all i EN. 3. For all terms a' we have [at]; <Ilo •

Lemma 1. [Nn];=O for all i EN.

Proofby induction on n. By definition [No]; = [0]; = o. Since Nn + 1 = [/' N n it follows from [[/'];=0 and [Nn];=O using clause 4 of the definition that [Nn+1];=0, too.

Lemma 2. (I) If aatba is not of the form JpNn then [aUt];~ [aatba]Jor all i EN. (2) For all i~g(1 we have [bU];~[iftba];.

Proof (I) follows immediately from the definition of [aatba];. (2) likewise follows from this definition if if'ba is not of the form JpNn. Otherwise (2) follows from Lemma I.

Lemma 3.

ifi~g(1,

Proof 1. Suppose aa'ba is not of the form JpNn. Then the assertion follows from the definition of [aatba];.


2. Suppose aUW' is a term JpNn • Then u=o and therefore gu=O and

if i=O,

and the assertion holds.

3. Estimates of the Order of Terms

By the order of a term a' we mean the ordinal term [a']o. We shall prove that b' has a smaller order than a' if a' reduces to b' in one step.

[a<] < [b'] means that [a']o < [b']o and [a']j ~ [b']j for all i ~gr.

Proof Since [Ku,]o= I we have

By Lemma 2(2) [a']j~[Ku,a']j foralli~gr.

By Lemma 2(1) [Ku~'l~[Ku~'bU]j for all iE N. It follows that [a']o<[Ku,a'bU]o and [a']j~[Ku~'bU]j for all i~gr.

Proof. For brevity we put aj:=[aPU'l, bj:=[bPUl and cj:=[cPl. From the definition and Lemma 3 we have

~j :=[apa'CP]j~{2~;+1 x (aj*,cj) if i~gp, a j if i>gp

"=[bpa P].:!({2~;+1 x (bj*,cj) ifi~gp, 1/, . C, "'" b. , ffi>gp

(j : = [apa'cP(bpacP)l = {:~j; + 1 X (~j *' 1/j) if i ~gu, .. if i>gu.

Spa, has type (pur)(pu)pr which has degree

max(gp + 2, gu + 2, gr + I).

apa, has type pur which has degree

max(gp + I, gu + I, gr).

16. Orders of Terms 109

bPtI has type P(1 which has degree

max(gp+ 1, g(1).

By. definition we therefore have

{201i + 1 x (1 # a;) ifi~max(gp + I,g(1+ I,g-r)

a.i : = [SPtI.aPtlt]i= 1 if i=max(gp +2, g(1+2, g-r + 1) o if i> max(gp +2, g(1+2, g-r + 1)

p .. = [S aPtltbPtll = {2fli + 1 x (a. 1 #b;) if i~max(gp + 1, g(1) •. Ptlt • a.i if i>max(gp+ I,g(1)

._ ptltptlP _{2Yi +1X(Pi#C;) ifi~gp Yi,-[Sptlta b C ]i- R 'f'

Pi 1 l>gp.

(1) lfmax(gp+ 1, g(1)<i~max(gp+ 1, g(1+ I,g-r) then ei<Yi'

Proof Here 'i=ei~ai and Yi=Pi=a.i=21ZI+l x (1 #ai»ai' and therefore 'i<Yi'

(2) lfgp<i~max(gp+I,g(1) then

ei#'1i#'i#2~2'i+l#1 x (1 #ai#b;)

and

Further

Proofby induction on max(gp+ I,g(1)-i. 1. Supposegp+I~g(1=i.

Then by (1) 'i+ 1 <Yi+ 1 and so 'i+ 1 # 1 ~Yi+ 1 and 2'1+ 1 # 1 x (1 #ai#bi) ~2Yl+l x (1 # ai # b;) where the assertion follows by (2).

2. Suppose g(1 <gp + 1 = i. Then a.i+1 ~O, 'i=~i~ai and '1i~bi and so

ei#'1i#'i#2~(21 x (1 #ai» #bi~(2011+1 x (l #ai»#bi =a.i#bi~2fli+l x (a.i#bi)=/Ji=Yi'


3. Suppose gp < i<max(gp + I, gO').

Then by induction hypothesis ej+ 1 *'1j+ 1 * Cj+ 1 *2~Yj+ 1 and so Cj+ 1 * I <Yj+ 1

whence the assertion follows by (2). 4. Suppose i ~gp.

Thenej~2~i+ I x (a j * Cj),'1j ~2q'+I x (bj*cj),Cj~2"+ I x (ej *'1J,(Xj=2~i+ I x (I *a j ),

Pj=2f1·+ , x «(Xj*bj) and Yj=2Yi +1 x (pj*cJ. It follows that

and

The assertion now follows on using the induction hypothesis ej+ 1 * '1j+ 1 * Cj+1 *2~yj+l'

It follows from (I) and (3) that ej < Yj for all i~!JT and therefore [aPt1tc"(II"'cP)] < [S Pt1taPt1t bPt1 c"].

Lemma 6. [b t ] < [JtOa"b1.

Proof By Lemma 2(1) [JtOa«Jo~ [JP]o= 1. It follows that

By Lemma 2(2) if i~gr: then [bt]j ~ [JtOa"br:]i" This yields the assertion.

Proof For brevity we write a j : = [att]j and bj : = [bt]j' By definition and Lemma 3 we have

if i~gr:+ I, if i=gr:+2, if i>!JT+2

Y, .=[ ''(IN "bt)],~ x a j '1j 1 l ..... gr: {2"+' ( * ) 'f'~

'o,' a t na ,..... 'f' aj 1 1>!JT

{2~i+' x (I *aJ if i~gr:+ I (Xj:=[JtNn+latt ]j= n+1 ifi=g-r+2

o if i>g-r+2

P,:=[J M a"bt].= x (Xj"11" j 1 l ..... gr:, {2f1• + I ( .u. b) 'f' ~ , t"'n+l '(Xj if i>g-r.

16. Orders of Terms 111

We prove ~i<CXi and '1i'*:'i#I~,8i for i~g-r+l by induction ong-r+l-i. From this it follows that, in particular, 'i <,8i for all i ~g-r which yields the assertion.

1. Supposei=g-r+ 1, then~i=2n x (1 #a i )<2n + 1 x (1 #a;)=cxjand'7i#'i# 1 = ~i#ai# 1 =2n x (l #a;)#ai# 1 ~2n+ 1 X (l #a;)=cxi=,8i'

2. Suppose i~g-r. Then by the induction hypothesis ~i+ 1 < CXi+ l' ~i+ 1 # 1 ~CXi+ 1

and '7i+ 1 #'i+ 1 # 1 ~,8i+ l' It follows that

~i=2~i+ 1 x (l # a;) < 2"i + 1 x (1 #ai)=cxi

and

'7j#'i~'7i#(2'i+1 x (ai#'7;»~(2'i+1 x a i )#(2'i+I#1 x '7i) =(2'i+1 x a;)#(2~i+l#"+I#l x (~i#b;» =(2"+ 1 X a;) #(2~i+l#~i+1 #'i+l#1 X (l #ai» #(2~'+I#"+I# 1 X b;).

Therefore

'7i#'i# 1 ~(2~i+l#~i+I#'1+1#2 x (l #a;»#(2~i+I#'i+l#1 x b;)

and by the induction hypotheses

'7i#'i# 1 ~(2"1+1#1Ii+1 x (l *ai»#(2I1i +1 x b;) =2I1i + 1 X «2"1+ 1 x (l #ai»# b;) =2111+1 x (cxi #b i )=,8;-

Lemma 8. [b"] < [c"] implies [a"tb"] < [a"tc"].

Proof By Lemma 1, [b"] < [c"] implies that c" is not a numeral. Therefore a"tc" is not of the form JpNn • It follows from the definition and Lemma 3 that

[aatba).:!( " {2Ia'''bO'Ii+ 1 X ([a"t]. # [ba ].) if i~gu

,,,,, [aat]i if i> gu

at a _{2IaO"CO'li+lx([aat]I#[Ca];) ifi~gu [a c ]i- [aat]; if i>gu.

By hypothesis [ba]j ~[C1i for all i~gu. It follows by induction on gu-i that [aatba]i~[a"tca]i for all i~gu. Hence [aatba]i~[a"tca]i for all ie N. Using the hypothesis [ba]o<[ca]o it also follows that [aatba]o<[aatca]o' This yields the assertion.

Lemma 9. If bat is not an iterator then [aat] < [bat] implies [aatca] < [batca].

Proof In this case by the definition and Lemma 3 we have

[a"t ca ] . :!( " {2IaO"CO'II+ 1 X ([aat]. # [ca].)

,,,,, [aat];

[batca).= " {2IbO"CO'li+ 1 X ([bat]. # [ca].)

, [b"t];

if i~gu,

if i>gu

if i~gu

ifi>gu.


By hypothesis [a"t];~ [b'7t l for all i ~max(gO"+ I, g"). By induction on i it follows that [a"te"]; ~ [b"te"]; for all i ~g(J. Hence [a"te"l ~ [b"te"l for all i ~g". Using the hypothesis [a"t]o < [b"t]o it also follows that [a"te,,]o < [b"te"]o.

Theorem 16.1. If a~o 1>1 a~o then a~o has a smaller order than ala.

Proof Suppose ala 1>1 a~o. We prove [a~o] < [ala] by induction on the length of the term ala. This will yield the assertion.

There are two cases to consider when ala I> 1 a~o : 1. For i= I, 2, afo is a term bfd~\ ... , d~m where" = "I' ... , "m"o (m ~o). This

gives the following four subcases for bl, b~ : 1.1. bl is K"tatb" and b~ is at, 1.2. bl is Sp,,~P"tbpaeP, and b~ is aP"teP(bP"eP), 1.3. b~ is JtOatW and b~ is b" 1.4. bl is JtNn+ Jattbt and b~ is att(JtNnattbt). 2. For i= I, 2, aiD is a term a"tbf d1T!, ••• , d:,m where"="I ... "n"o (n~O) where

a"t is irreducible and b~:} 1 b~. In the first case [b~] < [bl] holds by Lemmata 4-7. Then by Lemma 9 it follows

that [a~o]<[a~o]. In the second case [bn<[bn by the induction hypothesis. Then [Q:(Jtb~] < [a"tbn by Lemma 8 and [a~o] < [a~o] by Lemma 9.

Theorem 16.2. Every term has a normalform.

Proof This follows at once from Theorem 16.1 by induction on the ordering relation of the ordinal terms < eo, in other words by transfinite induction up to eo.

4. Equality of Terms

Definition. If a", bt are terms then a" = bt means that a" and bt have the same normal form.

Obviously this equality relation is an equivalence relation such that

a" =bt => 0"=" all> bt => at = bt.

Theorem 16.3. at=bt implies e"(::) = e"(!:}

Proof By hypothesis at and bt have the same normal form dt. It is therefore sufficient to prove:

We prove this assertion by induction on n.

17. The Formal System FT of Functionals of Finite Type

I. Suppose n = O. Then at and dt are identical and the assertion is trivial. 2. Suppose n>O. Then there exists a~ such that

113

By Theorem 16.2 CO"(::) has a normal form eO". Therefore there exists mE 1\1

such that

By Theorem 15.2 it follows that

Remark. It follows from 16.3 that equality for terms is compatible with arbitrary reductions. That is: In order to reduce a term we do not need to use the uniquely determined reduction procedure given above but can carry out the reduction of a term at at any place in at without altering the normal form.

§17. The Formal System FT of Functionals of Finite Type


Inductive Definition of the formulas of the system FT: 1. If at and bt are terms of the same type then at=bt is aformula. 2. If A and B are formulas, then (A - B) is also aformula. We use A, B, C, D, E, F, G, ~[at], ~[at] (possibly with subscripts) as syntactic

variables for formulas where ~ and ~ are the corresponding nominal forms. We use r, r i , L1, L1i to denote finite (possibly empty) sequences of formulas.

As in §6 we use the abbreviation:

for


If r is the sequence Ai' ... , An (n ~ 1) then r -+ B denotes the formula

Ai-+ ... -+ An-+ B.

If r is empty then r -+ B denotes the formula B. r c .1 means that every formula in the sequence r is also a formula in the

sequence .1.

Inductive Definition of the zero functional Ot of type 't. l. 0°:=0 2. OC1t: = KC1tOt

2. Deduction Procedures

The axioms of the system FT are

(Ax. K) KC1tatbC1 =at (Ax. S) SpC1taPC1tbPC1cP = aPC1tcP(bPC1cP) (Ax. JO) JtObttct = ct (Ax. JY) Jt(YaO)bttct=btt(J~Obttct) (Ax. =) at = bt -+ ~[atJ -+ ~[bt].

The basic inferences of the system FT are:

(Str) r-+BI-L1-+B, ifrcLi. (Cut) r-+ A, A-+ BI- r-+ B

{~[OJ, ~[xoJ -+ ~[YxoJ I- ~[aoJ, (CI) provided the variable XO does not occur in the nominal form~.

The basic inferences (Str) and (Cut) are the structural inference and the cut corresponding to those in §6. (CI) is the inference rule of complete induction.

Inductive Definition of j-!!B ("B is deducible with order n") for n EN: 1. If B is an axiom then j-!!B. 2. If A I- B is a basic inference (Str) and j-!!A then j-!!+ 1 B. 3. If Ai' A2 I- B is a basic inference (Cut) or (CI) and j-!!iA; (i= 1, 2) then j-!!B

where n: =max(n i , n2)+ l. I- B ("B is deducible in the system FT") means that there exists n E N such that

p!- B.

Theorem 17.1 (Substitution rule) . .lfj-!!~[xtJ, where the variable xt does not occur in the nominalform~, then j-!!~[atJfor every term at of type 'to

Proofby induction on n. 1. Suppose ~[xtJ is an axiom. Then ~[atJ is also an axiom. 2. Suppose j-!!~[xtJ was obtained by a basic inference (Str) or (Cut). Then by

17. The Formal System FT of Functionals of Finite Type 115

the induction hypothesis the formulas obtained from the premisses by substituting at for xt are deducible with the same orders as the premisses. Using the appropriate basic inference (Str) or (Cut) we obtain f-!!ff[at ].

3. Suppose f-!!ff[xt] was obtained by a basic inference (CI). Then the formula ff[xt] is a formula ~[bO] and we have

where n=max(n1, n2 )+ 1 and where yO does not occur in~. We choose a variable z" different from xt which occurs in neither ~ nor in at. By I.H. (induction hypothesis) it follows that

Let ~* be the nominal form obtained from ~ by substituting at for xt. By I.H. it follows that

Using a (CI)-inference it follows that

b~t ~* [bo(::) ] is the formula ff[at ].

3. The Consistency of the System FT

Definition. A formula is said to be closed if it contains no variable.

Inductive Definition of truth andfalsity for a closed formula.

1. A closed equation at = bt is true if the functionals at and bt have the same normal form. Otherwise the equation isfalse.

2. A closed formula A --+ B is true if A is false or B is true. Otherwise the formula isfalse.

Lemma 1. If two funetionals at and bt have the same normal form then a closed formula ff[at ] is true if, and only if, the formula ff[bt ] is true.

Proofby induction on the number of symbols --+ occurring in ff[at ]. 1. Suppose ff[at ] is an equation. Then there are terms e", d" and a variable

xt such that ff[at ] is the equation

(1) ell(::)=dll (::)


and .F[bt ] is the equation

By Theorem 16.3 the functionals

have the same normal forms as the functionals

Therefore (1) is true if, and only if, (2) is true. 2. Suppose .F[at ] contains the symbol-.. Then the assertion follows from

the induction hypothesis and the truth definition.

Theorem 17.2. Every deducible closed/ormula is true.

Proof Suppose C is a closed formula and ~c. We prove by induction on n that C is true.

I. Suppose C is an axiom (Ax. K), (Ax. S) or (Ax. JO). Then C is an equation ct = d t for which ct t> 1 d t holds. It follows that C is true.

2. Suppose C is an axiom (Ax. J.9):

By Theorem 15.1 the normal form of aO is a numeral Nm • Then by the reduction procedures the equation

is true. Hence by Lemma I C is true. 3. Suppose C is an axiom (Ax. = ):

at = bt -. .F[at] -. .F[bt ].

If at = bt is false, then C is true. If at = bt is true then by Lemma I it follows that .F[at ] is true if, and only if, .F[bt ] is true. Thus in every case C is true.

4. Suppose ~C was obtained by a basic inference (Str). The premise of this inference is a closed formula which is true by I.H. It follows that C is true by 2. of the definition of truth.


5. Suppose f-!!C was obtained by a basic inference (Cut). Then C is a formula r - B and we have

f-!!tr- A and f-!!2A- B

where n=max(n 1 , n2 )+ 1. Let A* be the formula obtained from A by replacing each variable x< occurring in A by the zero functional of the same type. Then by Theorem 17.1 it follows that

f-!!tr_A* and f-!!2A*-B.

r - A* and A* - B are closed formulas and therefore are true by I.H. It follows that r - B is true.

6. Suppose f-!!C was obtained by a basic inference (CI). Then C is a formula ff[a O ] and we have

where n = max (n l' n2 ) + 1. By Theorem 17.1 it follows that for all mEN

By I.H. the closed formulas ff[O] and ff[Nm] - ff[Nm+ 1] are true. It follows by induction on m that ff[N m] is true for every numeral N m. Now the normal form of a,o is a numeral by Theorem 15.1. So by Lemma 1 it follows that ff[a O ] is true.

Remark. The converse of Theorem 17.2 does not hold in general. Thus, for example,

is a true closed formula which is not deducible in FT. However, every true closed equation is deducible in FT. This follows from the fact that every equation a< = b< such that a< r> l b< holds is deducible in FT.

Corollary to Theorem 17.2. The formal system FT is syntactically consistent. That is: Not every formula of the system FT is deducible in FT.

Proof The false closed equation 9'0=0 is not deducible in FT by Theorem 17.2.

Remark. Besides elementary methods our consistency proof for the system FT uses only Theorem 16.1 which we proved by transfinite induction up to eo. (Theorem 16.1 was used to prove Theorem 16.3 and hence also Lemma I which we used to prove Theorem 17.2.)

4. Fundamental Deduction Rules

Theorem 17.3. f- A-A.

118

Proof KooOO=O and

VI. Functionals of Finite Type

are (Ax. K) and (Ax. = ) axioms (where in the (Ax. = ) axiom!F is a I-place nominal form A in which the nominal sign * 1 does not occur). Using a cut we have f- A -> A.

The next theorem shows that equality is an equivalence relation which is compatible with the mapping of terms and formulas.

Theorem 17.4 (Rules for equality).

(= 1) f-!F[a t ] implies f-at=b t -> !F[bt ] (=2) f-at=at

(=3) f-at=bt -> bt=at

(=4) f-at=bt -> bt=ct -> at=ct

(=5) f-at=bt -> ca(::)=caC:}

Proof 1. Using a structural inference on the (Ax. =)

at = bt -> !F[at ] -> !F[bt ]

we obtain

!F[at ] -> at = bt -> !F[bt ].

Then using f-!F[a t ] by a cut we have

f-at = bt -> !F[bt ].

2. By (= 1) from the (Ax. K)

KoptO=at

we obtain

KoptO =at -> at =at.

Using a cut we have f-at=at. 3. (= 3) follows from (=2) by (= 1). 4. Using a structural inference on the (Ax. =)

bt = ct -> at = bt -> at = ct

we obtain (=4).


5. By (=2) we have

Hence by ( = 1) we obtain ( = 5).

Theorem 17.5. lfl-F[O) and I-F[9'x O) where the variable XO does not occur in the nominal form F then I-F[aO) for every term aO of type o.

Proof By a structural inference from I-F[9'x O) we obtain

I-F[xO) -+ F[9'xO).

Now a (CI)-inference using I-F[O) yields I-F[a°).

Corollary to Theorem 17.5. l-ao=O-+ A and l-ao=9'x°-+ A yield I-A provided XO occurs in neither aO nor A.

Proof By Theorem 17.5 from the hypothesis we have l-aO=a°-+ A. Hence by (=2) we have I-A.

Definition of terms for definition by cases.

D,[aO, b', c') : =J,aO(K..c')b' v[aO) : =JoaO(KOo.°)(9'O)

Corollaries. The following equations are deducible

(DO) D ,[0, b', c') = b' (D9') D,[9'ao, b', c')=c' (vO) v[O) =9'0 (v9') v[9'aO) =0.

Proof By (Ax. JO), (Ax. J9') and (Ax. K) we have

D,[O, b', c') = J,O(Knc')b'=b' D,[9'~, b" c')=J,(9'aO)(Knc')b'=K.,c'(J,aO(K..c')b')=c'.

(vO) and (v9') are special cases of (DO) and (D9').

Theorem 17.6. l-9'ao=O-+ A.

Proofby induction on the number of arrows which occur in the formula A. 1. Suppose A is an equation b' = ct. By ( = 5) we have


Using (DsP) and (DO) we obtain

f-sPao=O -+ b<=c<.

2. Suppose A is a formula B -+ C. Then by I.H. we have

f-sPao=O-+ C.

Using a structural inference we obtain

5. Addition and Multiplication

Definitions.

aO + bO : =Jobo sPaO aO. bO : =JobO(Joao SP)O.


(+0) aO+O=ao (+sp) aO+sPbo=sP(ao+bO) (·0) aO·O=O (.sp) aO ·sPbO =aO ·bo +ao.

Proof By (Ax. JO) and (Ax. JsP) we have

aO + 0 = Jo0sP aO = aO aO + sP bO = Jo(sP bO)sP aO = sP( Jobo sP aO) = sP(aO + bO)

aO·O=JoO(Joa°sP)O=O aO. sPbo = Jo(sPbO)(Joa°sP)O = Joao sP(Job°(Joa°sP)O) =aO. bO + if

Remark. All the equations for the basic properties of addition and multiplication follow from the recursion equations (+ 0), (+ sP), (·0) and (. sP) by means of (CI)-inferences:

aO + (bO + cO)=(aO +bO) + CO aO . (bO + CO) = aO . bO + aO . CO

aO·(bo·cO)=(ao·bO)·co aO+bo=bo+ao

aO. bO = bO. aD.


6. The Identity Functional It and A-Abstraction

Definition.

Corollary. It is afunctional of type 't! and I-Itat=at.

Proof By (Ax. S) and (Ax. K) we have

Inductive Definition of hP(c t ).

1. AXP(XP): = Ip. 2. If x P does not occur in ct then set

3. If x P in a'''bt1 then set

Corollary. AXP(Ct ) is a term of type po in which the variable xP does not occur.

Proofby induction on the length of the term ct.

1. AxP(xP)dP=I dP=dP=Xp(dP). P xP

2. If x P does not occur in c" then we have

3. If x P occurs in at1tbt1 then we have

Using the induction hypothesis we obtain

121


Notation. In the sequel we write

for

AXrl( .. . (AX:"( c'» ... ).

xr', ... , x:" must always be distinct variables. If these variables do not. occur in ar', ... , a:" then by Theorem 17.7 we have

7. The Predecessor Functional and the Arithmetic Difference

Definition of an auxiliary functional.

U is a functional of type (00)00. Applying Theorem 17.7 we have

Using (Ax. JO) and (Ax. JY') we obtain

(1) I-UaooO=aOO(Y'O) (2) I- UaOO(Y'O) = Y'(aOO(Y'O».

Definition of the predecessor functional

P is a functional of type 00. By Theorem 17.7 we have

Using (Ax. JO) and (Ax. JY') we obtain

(3) I-PO=KooOO=O I-P(Y'CO) = U(JoocOU(KooO»O.

Using (1) we obtain


Theorem 17.8. For the predecessor functional P we have:

(PO) I-PO=O (P9') I-P(9'aO) =aO.

Proof (PO) holds by (3). By (4) we have P(9'O) = JooOU(KooO)(9'O) = KooO(9'O) = 0 and therefore

(5) I- P(9'O) = o.

By (4) we also have

Using (2) and (4) we obtain

Hence

By a (CI)-inference from (5) and (6) we obtain

Proof By ( = 5) we have

The assertion now follows using (P 9').

Definition of the arithmetic difference a°..:...bo.


(..:...0) a°..:...O=aO (..:...9') a°..:...9'bo = P(a°..:...bO).

Proof By (Ax. JO) and (Ax. J9') we have

a°..:...O=JoOPao=ao a°..:...9'bo =Jo(9'bO)PaO = P(Jobo PaO) = P(a°..:...bO).

124

Theorem 17.10. Thefollowing equations are deducible

(1) O..:..bo=O (2) Y'if..:..Y'bo=a°..:..bo (3) Y'a°":"ao=Y'O (4) a°":"ao=O.

Proof. 1. We have

(1.1) f-O..:..O=O

(1) follows from (1.1) and (1.2) by a (CI)-inference.

VI. Functionals of Finite Type

2. Now Y'a°..:..Y'O=P(Y'a°..:..O) = P(Y'aO) =ao=if":"O, therefore

(2) follows from (2.1) and (2.2) by a (CI)-inference. 3. We have

(3.1) f-Y'O..:..O=Y'O

(3) follows from (3.1) and (3.2) by a (CI)-inference. 4. From (2) and (3) we obtain a°":"ao =Y'if..:..Y'if=P(Y'a°..:..aO)=P(Y'O) =0,

and therefore

8. The Recursor

Definition of an auxiliary functional.


Qt is a functional of type (on)(o-r)01:. By Theorem 17.7 we have

Using (P9'), by Theorem 17.8 we have

Definition of the recursor Rt •

R t is a functional of type -r( on)01:. By Theorem 17.7 we have

Theorem 17.11. (Primitive recursion for type -r).

(RO) I- Rtat bOttO = at (R9') I- Rtatbott(9' CO) = bottcO(RtatbottcO).

Proof 1. From (2) using (Ax. JO) and (Ax. K) we have

Hence (RO) holds. 2. Using (Ax. J9') we obtain from (2)

RptbOtt(9' CO) = 'Ot(9' CO )(Qtbott )(KOta)(9' CO) = QtbOtt( 'OtcO(Qtbott)(KOtat»(9'cO).

From (I) and (2) we obtain

Hence (R9') also holds. This theorem shows that RtatbOtt is a termpt depending on at and batt for which

the primitive recursion equations

fOtO=at fot(9' CO) = bottcO(fOtcO)

of type -r are deducible.


9. Simultaneous Recursion

Theorem 17.12 (Simultaneous n-fold primitive recursion). Given terms a? and b't,···fnf'(i= 1, ... , n) there are terms hOt; sueh that the equations

hOf'O=aI; hOf;([/eO) = brf, .. . fnf;eO(ft'eO) ... (J..0fneO)

are deducible for all i= 1, ... , n and all terms eO of type o.

Proofby induction on n. If n = 1 the assertion holds by Theorem 17.11 if we set

Now suppose n> 1. For brevity we write bi' e, dfor brf' ... fnf;,~, dO. Let

(J : = (o't 1) .•• (otn - 1)'tn

S" :=.A.x~f' ... x::-il(a!n) to"" : =A.XOy"zf' ... Z:':i '(bnxO(zf'xO) .. . (Z:':i 'xo )(y"zf' ... Z:':1- I»

9 :=R"s"tO"",

where the variables xf', ... , x:':i' do not occur in a!n and the variables xO, y", Z~f', ••• , z::-i' do not occur in bn. Then 9 is a term of type o(o't1) ••• (O'tn_ 1)'tn such that the following hold by Theorem 17.11:

I-gO=s" I-g([/ e) = to"" e(ge).

By the definitions of s", to"<I and Theorem 17.7 we have

(1) I-gOe~f' ... e::-i' =a!n (2) I-g([/e)e~f' .. . e:':i' = bne(e~f'e) ... (e:':ile)(gee~f' .. . e:':i').

For i = 1 , ... , n - 1 let

Vi is a term oftype o(o't 1) ..• (o'tn- 1)oti • By the induction hypothesis there are terms hi of type Oot i (i = 1, ... , n - 1) such that the equations

hiO=ur f ;

hi([/ e) = vie(h1 e) ... (hn- 1 e)

17. The Formal SystemFT of Functionals of Finite Type 127

are deducible for all i= I, ... , n-l and all terms c of type o. From the definitions of uf t " Vi and Theorem 17.7 we have

(3) I-hiOd=af'. (4) I-h;(!7c)d= Dt,[d~c, hiCd, bic(h1 cc) ... (hn - 1 cC)(gC(hl c) ... (hn- 1 c»].

By (~O) and (0!7) from (4) we obtain

(4.1) I-d~c=O~ hi(!7c)d=hicd. (4.2) I-d~c=!70 ~ hi(!7c)d=bAh l cc) ... (hn_1cc)(gc(h 1 c) ... (hn- 1 c».

Now we define

/;: =.hO(hiXOxO) (i= I, ... , n-I) In: =h°(gxO(hlXO) ... (hn_lXO».

Then/; is a term of type 07:i' By Theorem 17.7 we have

(5) {1-J;C=hiCC (i=I, ... ,n-l) I-Inc =gC(hl c) ... (hn- 1 c)

From (3), (4.2) and (5) we have, for i= I, ... , n-I,

since by Theorem 17.10 (3) l-!7c~c=!70 holds. From (I) and (5) we obtain

By (4.1) we have

Using (2) we obtain

) I-{XO~c=o~ g(!7xO)(hl(!7c» ... (hn_l(!7c» (8 . = bnxO(hl CXO) ... (hn- 1 cxO)(gXO(hl (!7 c» ... (hn- 1 (!7 c»)

From !7xo~c=O by Theorem 17.10 we obtain

Hence from (8) and (2) we obtain

{(XO~c=O ~ gxO(h 1(!7c» ... (hn_ 1(!7c»=gxO(h l c) ... (hn- 1 c»

I- ~ (!7xo~c =0 ~ g(!7xO)(hl (!7c» ... (hn - 1 (!7c) =g(!7XO)(hl c) ... (hn- 1 c»


From (1) we also have

Using a (CI)-inference we have

Using I-c...!...c=O (by Theorem 17.10) and (8) we obtain

Hence by (5) we have

With (6), (7) and (9) the proof of the theorem is complete.

Theorem 17.13 (Induction Theorem). For i= 1, ... , n let bi be a term of type o'tl ••. 'tn't i . Let §Txo,Yl', ... ,y~"] be aformula with distinct variables xO,yl', ... ,y~n which do not occur in the nominal form § nor in the terms b l , ... , bn. Suppose

(I) I- ff[O, yl',.··, y~"] (II) I-ff[xO, blxOyl' ..• y~n, .. ~, bnxoYl' ... y~"] ~ ff[9"xo, yl', ... , y~"]

hold. Then I-ff[aO, aI', ... , a~"] holdsforall terms aD, aI', ... , a~" of types 0, 't l , .•. , 'tn.

Proof By Theorem 17.12 for the terms if and aI', ... , a~" there are terms.!; of type O'ti such that for all i= 1, ... , n and all terms CO of type 0 the equations

(1) f;O=aIi (2) .!;(9"cO) = bi(a°...!...9" CO)(fl CO) ... (fncO)

are deducible. It follows from a°...!... ZO = 9"xO that

Therefore, substituting.!;zo for yIi, hypothesis (II) and the equations (2) yield

(3) I-a°...!...zo = 9"xo ~ ff[a°...!...9"z°,fl (9"zO), ... ,f,,(9"zO)] ~ff[a°...!...zo,flZo, ... ,fnzO].

Also by hypothesis (I) we have

(4) I-a°...!...zo = 0 ~ ff[a°...!...9"z°,fl (9"zO), ... ,f,,(9"zO)] ~ §[a°...!...zD,flZo,···,fnz°].


By Theorem 17.5 from (3) and (4) we obtain

f-ff[a°"":""Sf'z°,f&9'zO), ... ,f,.(Sf'zO)] -+ ff[a°....:....z°,f1zo, ... '/nzO].

Hence

(5) f- {(ff[ao....:....zo,fl Zo'. ··,fnzO] -+ ff[aO, a~l, ... , a~"]) -+ (ff[a°....:....Sf'z°,fl (Sf'zO), .. ''/n(Sf'zO)] -+ ff[aO, a~I, ... , a~"]).

Also from the equations (1) we have

Using a (CI)-inference (5) and (6) yield

Using f-a°"":""aO=O (by Theorem 17.10) and hypothesis (I) the assertion f-ff[aO,a~I, ... ,a~"] follows.

Theorem 17.14. Let ff[aO, bO] be aformula. Let xO, yO be distinct variables of type 0 which do not occur in the nominal form ff. Suppose

(I) f-ff[xO,O] (II) f-ff[O, yO] (III) f-ff[xO, yO] -+ ff[Sf'xO, Sf'yO]

hold. Then f-ff[aO, bO] holdsfor all terms aD, bO of type o.

Proof ZO = P(Sf' ZO) and hypothesis (III) yield

Also by hypothesis (I) we have

By Theorem 17.5 from (1) and (2) we obtain

The assertion now follows from (3) and hypothesis (11) by Theorem 17.13.

10. The Characteristic Term of a Basic Formula

Definition. Formulas which only contain equations between terms of type 0 are said to be basic formulas.


Inductive Definition of the characteristic term x[ C] of a basic formula C. 1. x[ao=bO] : = (a°...!..bO) + (b°...!..aO) 2. X[A -+ B] : = v[X[A]]- X[B]. By this definition X[ C] is a term of type 0 for every basic formula C. In order to prove the next Theorem 17.15, we need the following Lemmata:

Lemma 2. The following formulas are deducible

(1) O+bo=bo (2) O·bo=O (3) f/O·bo=bo (4) aO+bo=O-+ bO=O (5) aO+bo=O-+ aO=O.

Proof. 1. We have

(1.1) f-O+O=O


(2.1) f-O·O=O

(2.2) f-O·xo=O-+ O·f/xo=O.


(3.1) f-f/O·O=O

and f/O· f/xo = f/O· XO + f/O = f/(f/O· XO + 0) = f/(f/O· XO). Hence

(3) follows from (3.1) and (3.2) by a (CI)-inference. 4. Trivially


(4) follows from (4.1) and (4.2) by Theorem 17.5. 5. (5) follows from (4).

Proof By Theorem 17.10 and Lemma 2 we have

therefore

Similarly we obtain

By Theorem 17.10(2) we also have

The assertion follows from (lH3) by Theorem 17.14.

Theorem 17.15. Thefollowing holdfor every basic formula C.

(1) I-x[C] =0-. C (2) I-C-. X[C]=O.

131

Proofby induction on the number of -. symbols, occurring in the basic formula C. 1. Suppose C is an equation aO = bO.

By Lemma 2 from the definition of x[aO = bO] we have

I-x[ao=bO] =0-. a°..:...bO=O I-x[ao=bO] =0-. b°":"'aO=O.

By Lemma 3 and Lemma 2(1) we have


Hence we obtain

2. Suppose C is a basic formula A-B. Since X[A - B] =v[X[A]]· X[B], v[O] =9'0 and YO· X[B] =X[B] we have

(2.1) f-X[A]=O- X[A- B]=X[B].

By the induction hypotheses

we obtain

(2.2) f-X[A --+ B] =0- A-B.

The induction hypotheses f-X[A] =0- A and f-B- X[B] =0 together with the deducible formula

A - (A --+ B)--+ 1J

yield

f-X[A] =0- (A- B)- X[B]=O

by cuts. Hence using (2.1) we have

(2.3) f-X[A] =0 --+ (A - B) - X[A --+ B] =0.

It follows from X[A - B] =v[x[A]] 'X[B], v[9'xO] =0 and o· X[B] =0 that

Hence we also have

(2.4) f-X[A]=9'xo- (A- B)--+ X[A--+ B]=O.

By Theorem 17.5 frOin (2.3) and (2.4) we have

(2.5) f-(A - B) - X[A - B] =0.

(1.1), (1.2), (2.2) and (2.5) complete the proof of Theorem 17.15.

17. The Formal System FT of Functionals of Finite Type l33

Theorem 17.16.

holds for every basic formula A and every formula B.

Proof By Theorem 17.15 we have I-A -. X[A] =0. Hence I-x[A]=9"a°-' A-. 9"ao =O. The assertion follows by Theorem 17.6.

Chapter VII

Pure Number Theory

In §18 we develop a formal system of pure number theory which can, in particular, be easily interpreted in the system Ff of §17. The interpretation by K. GOdel [2] is carried out in §19.

§18. The Formal System PN for Pure Number Theory


We use the following primitive symbols: 1. Denumerably infinitely many free and bound number variables. 2. The symbols 0, f, +, " =, -+ and V. 3. Round brackets. Nominal forms are defined in the usual way and denoted by capital script

letters.

Inductive Definition of number terms: 1. The symbol 0 is a number term. 2. Every free number variable is a number term. 3. If t is a number term so is tf. 4. If sand t are number terms then so too are (s+ t) and (s· 1). For brevity we omit the outer round brackets. We regard· as stronger than +

and thus abbreviate, for example, «tl . t2) + t3) by tl . t2 + t3'

Syntactic symbols: u, Ui ' v, Vi for free number variables x, Xi' y, Yi' Z, Zi for bound number variables s, Si' t, ti for number terms.

Inductive Definition of formulas 1. If sand t are number terms then s=t is aformula. 2. If A and B areformulas so too is (A -+ B).

18. The Formal SystemPN for Pure Number Theory 135

3. If F[u] is a formula in which the bound variable x does not occur then IIxF[x] is also aformula.

We use A, B, C, D, E, F, G, IIxF[x], 1I~[x] (possibly with indices) as syntactic symbols for formulas.

For brevity we write

for

(Ai - ( ... - (An- B) ... ».

2. The Deduction Procedure

We choose the axioms and basic inferences of the system PN so that every formula of our language which is deducible in classical predicate calculus is also deducible in PN. In doing this we do not work from positive and negative parts of formulas as we did in the system CP for classical predicate calculus, but we choose the axioms and basic rules of inference in another way to facilitate the interpretation of PN in FT (§19).

~xioms for the system PN:

(AI) A-A (A2) IIxF[x] - F[t] (A3) u'=O- A (A4) u'=v' - u=v

(A5) u+O=u (A6) u+v'=(u+v)' (A7) u·O=O (AS) u·v'=u·v+u.

Basic Inferences for the system PN:

(BI) B I- A- B (B2) A - B ~ C - D I- A - C - B - D (B3) A - B - C, C - D I- A - B - D (B4) A - A - B I- A - B (B5) u=u- A I- A (B6) F[s] I- s=t- ~[t]

(B7) A - ~[u] I- A - IIxF[x] provided the free number variable u does not occur in the conclusion.

(BS) F[O], F[u] - ~[u'] I- ~[t] provided the free number variable u does not occur in the nominal form F.

136 VII. Pure Number Theory

(AI), (BI)-{B4) are the rules for positive implicationallogic, (BS), (B6) the rules for equality, (A2), (B7) the rules for V-quantification. The Peano axioms for arithmetic are given by (A3), (A4), (BS) and the recursion equations for addition and multiplication are given by (AS)-{AS). Hence from a logical point of view arithmetic is axiomatised as a fragment of intuitionistic predicate calculus. In this system PN we shall also obtain all the rules of classical predicate calculus for the language of PN.

As previously we shall use

to denote a permissible inference of the system PN and I-F to denote that F is deducible in the system PN.

3. Basic Properties of Deducibility

Lemma 1. A - B, B - C I- A-C.

Proof By (BI) from A - B we have A - A-B. By (B3) from B- C we have A - A - C. Hence A - C by (B4).

Lemma 2. A, A - B I- B.

Proof By (8 I) from A we have u = u - A. By Lemma I from A - B we have u=u- B. Hence B by (BS).

Lemma 3. ,g;;[v] I- Vx,g;;[x] provided the free number variable v does not occur in the nominal form ,g;;.

Proof By(BI)from,g;;[v] wehaveu =u - ,g;;[v]. By(B7)wehaveu=u- Vx,g;;[x]. Hence by (BS) we have Vx,g;;[x].

Theorem 18.1 (Substitution rule). ,g;;[u] I- ,g;;[t] provided the free number variable u does not occur in Ihe nominal form ,g;;.

Proof By Lemma 3 from ,g;;[u] we have Vx,g;;[x]. Using the (A2)-axiom Vx,g;;[x] - ,g;;[I] by Lemma 2 we obtain ,g;;[I].

Theorem 18.2 (Rules for equality).

(I) I-I=t (2) I-s=l- I=s (3) I-S=I- ,g;;[s]- ,g;;[I] (4) I-s= I - s' = I'.

18. The Formal System PN for Pure Number Theory 137

Proof 1. By (BS), from the (AI)-axiom,

u=u-+ u=u

we obtain I-u=u. By Theorem 18.1 we have I-t=t. 2. By (1) 1-8=8 and then by (B6) 1-8=t-+ t=8. 3. By (B6) from the (AI)-axiom '[8] -+ '[8] we have

1-8=t-+ '[8] -+ '[t].

4. By (I) 1-8'=8' and then by (B6) 1-8=t-+ 8'=t'.

Lemma4. I-(A-+ B-+ C)-+ B-+ A-+ C

Proof This follows from the (AI)-axiom

(A-+ B-+ C)-+ A-+ B-+ C

by (B2).

LemmaS. A-+ B-+ CI- B-+A-+ C.

Proof This follows from Lemma 4 by Lemma 2.

Lemma 6. I-A -+ B-+ A.

Proof By (BI) from the (A1)-axiom A -+ A we have

I-B-+A-+A.

The assertion now follows by Lemma S.

Lemma 7. I-A -+ (A -+ B) -+ B.

Proof This follows from the (A I)-axiom

(A-+B)-+A-+B

by Lemma S.

Lemma 8. A -+ B I- (B-+ C) -+ A -+ C.

Proof By Lemma 7

B-+(B-+ C)-+ C


is deducible. From this and A - B by Lemma I we obtain

A- (B- C)- C.

By Lemma 5 we have

(B- C)-A- C.

Lemma 9. ~[OJ, ~[u'J I- ~[tJ provided the free number variable u does not occur in the nominalform~.

Proof By (Bl) from ~[u'J we have

~[uJ - ~[u'].

From ~[OJ by (B8) we obtain ~[t].

4. Properties of Negation

Definition: The negation of A, -, A, is the formula A - 0' = O.

Lemma 10. I-A - -, -,A.

Proof This follows from Lemma 7 by the definition of negation.

Lemma 11. I--,A - A-B.

Proof By the definition of negation

is an (A 1 )-axiom. By Theorem 18.1 from the (A3)-axiom u' = 0 - B we obtain

1-0'=0- B.

By (B3) we have

I--,A- A- B.

Lemma 12. A - B I- -,B_ -,A.

Proof This· is a special case of Lemma 8 because of the definition of negation.

Lemma 13. A- B- CI- A- -,C- -,B.

18. The Formal System PN for Pure Number Theory

Proof By Lemma 10 we have

f-C-+ -,C-+ 0'=0.

By (B3) from A -+ B-+ C we obtain

A-+ B-+ -,C-+ 0'=0.

by (B2) we have

Lemma 14. f--, -, s= t -+ s= t.

Proof By Theorem 18.2 (1) we have f-O=O. Hence by (Bl) we have

(1) f--,-,O=O-+ 0=0.

By Lemmata 11 and 5 we have

f--,u'=O-+ -,-,u'=O-+ u'=O.

By Lemma 2 using the (A3 )-axiom -, u' = 0 we have

(2) f--,-,u'=O-+ u'=O.

From (1) and (2) by Lemma 9 we obtain

f--, -,u=O-+ u=O.

Hence by Lemma 3

(3) f-'v'x(-' -,x=O-+ x=O).

By Theorem 18.2 (2) we have

f-O=v' -+ v' =0.

Using the (A3)-axiom

v'=O-+ 0'=0

by Lemma 1 we obtain

f--,O=v'.

139


By Lemmata 11 and 5 we have

I--,O=v' -+ -, -,O=v' -+ O=v'. By Lemma 2 we have

1--, -,O=v' -+ O=v'.

By (Bl) we obtain

(4) I-'Vx(-' -,x=v-+ x=v) -+ -, -,O=v' -+ O=v'.

By two applications of Lemma 12 we obtain

I--,-,u'=v'-+ -,-,u=v

from the (A4)-axiom

u'=v'-+ u=v.

By Lemma 8 we obtain

I-(-,-,u=v-+ u=v)-+ -,-,u'=v'-+ u=v.

By Theorem 18.2 (4) we have

I-u=v-+ u'=v'.

By (B3) we obtain

I-(-,-,u=v-+ u=v)-+ -,-,u'=v'-+ u'=v'.

Using an (A2)-axiom by Lemma 1 we obtain

(5) I 'Vx(-' -,x=v -+ x=v) -+ -, -,u' =v' -+ u' =v'.

From (4) and (5) by Lemma 9 we have

I-'Vx(-' -,x=v-+ x=v) -+ -, -,u=v' -+ u=v'.

Hence by (B7) we have

(6) I-'Vx(-' -,x=v -+ x=v) -+ 'Vx(-' -,x=v' -+ x=v').

By (B8) from (3) and (6) we obtain

I-'Vx(-' -,x=t-+ x=t).


The assertion

1--, -,s=t-+ s=t

now follows by Lemma 2 using an (A2)-axiom.

Lemma 15. I--,-,C-+ C.

Proofby induction on the number of -+ and V symbols occurring in the formula C. 1. Suppose C is an equation s= t. Then the assertion holds by Lemma 14. 2. Suppose C is a formula A -+ B. By Lemma 7 we have

I-A-+ (A-+ B)-+ B.

By two applications of Lemma 13 we obtain

I-A -+ -, -, (A -+ B) -+ -, -, B.

From the induction hypothesis

I--,-,B-+ B

by. (B3) we have

I-A -+ -, -, (A -+ B) -+ B.

Hence by Lemma 5 we have

1--, -,(A -+ B) -+ A -+ B.

3. Suppose C is a formu~a Vx.F[x]. Let u be a free number variable which does not occur in C. By two applications of Lemma 12 we obtain

1--,-,Vx.F[x]-+ -,-,.F[u]

from the axiom

Vx.F[x] -+ .F[u].

Using the induction hypothesis

1--, -,.F{u] -+ .F[u]

we obtain by Lemma 1

1--, -,Vx.F[x] -+ .F[u].


Hence by (B7) we have

Lemma 16. -,A - B I- -,B- A.

Proof By Lemma 12 from -,A - B we obtain -,B- -, -,A. By Lemma 15 the formula -, -, A - A is deducible. Hence by Lemma 1 we have -, B - A.

Lemma 17. A - B, -,A - B I- B.

Proof By Lemma 16 from -,A- Bwe have -,B- A. By Lemma 1 usingA- B we obtain -, B - B. By Lemma 10 the formula B - -, -, B is deducible. Hence by Lemma 1 we have -,B- -, -,B. Hence by (B4) we obtain -, -,B. By Lemma 15 -, -,B- B is a deducible formula. Hence by Lemma 2 we obtain B.

Lemma 18. -,A- C, B- CI- (A- B)- C.


By Lemma (B3) using B - C we obtain

A- (A-'.B)- C.

From -,A - C by (Bl) and Lemma 5 we obtain

Hence by Lemma 17 we obtain

(A- B)- C.

s. Positive and Negative Parts of Formulas

We define positive and negative parts of formulas, P-forms, N-forms and NPforms as in §3 (p. 20) except that the formula 0' =0 now plays the role of ..L. As before we use the following syntactic symbols: rJ', rJ'j for P-forms, %, %j for N-forms, fl, flj for NP-forms.

Every P-form is a nominal form *1 or rJ'o[C- *1] or %o[-,*tJ and every N-form is a nominal form rJ' o[ *1 - C] by the inductive definition given above on p. 20. Now rJ'0 and %0 are shorter nominal forms than rJ' and % respectively.

18. The Formal System PN for Pure Number Theory .143

Consequently we prove our theorems about P-forms and N-forms by induction on the length of these nominal forms.

Lemma 19. (1) f-A----.&'[A] (2) f-,A----. '%[A].

Proo/by induction on the lengths of &' and .%. 1. Suppose &' is *1' Then A ----. &,[A] is the (Al)-axiom A ----. A. 2. Suppose &' is &' o[ C ----. *1]. By the I.H. (induction hypothesis) we then have

f-(C----. A)----. &,[A].

By Lemma 6 the formula

is deducible, hence by Lemma 1 we have

f-A----. &,[A].

3. Suppose &' is .%0['*1]. By I.H. we then have

By Lemma 10 the formula A ----. , ,A is deducible, hence by Lemma 1 f- A ----. &,[A]. 4. Suppose.% is &'0[*1 ----. C]. By I.H. we then have

f-(A ----. C) ----. '%[A].

By Lemma 1 the formula

,A----.A----.C

is deducible, hence by Lemma 1 we obtain

f-,A ----. '%[A].

Lemma 20. (1) &,[A] f- '&,[B]----. A (2) '%[A] f- A ----. '%[B].

Proo/by induction on the lengths of &' and.%. 1. Suppose &' is * l' Then the assertion holds by (B 1). 2. Suppose &' is &'0 [C ----. * 1]' Then C is a negative part of &'[ B]. By Lemma 19

we therefore have



By the I.H. from &,[A] we obtain

-'&,[B] -+ C-+ A.

By Lemmata 5 and 1 we obtain

-,&'[B] -+ -,&'[B] -+ A.

By (B4) we obtain -'&,[B] -+ A. 3. Suppose &' is .IV 0[-' *1]. By the I.H. we then obtain from &,[A]

-, A -+ &'[ B].


-'&,[B] -+ A.

4. Suppose.IV is &'0 [ *1 -+ C]. Then C is a positive part of .IV[B]. By Lemma 19 we therefore have

I-C-+ .IV[B].

By I.H. from .IV[A] we obtain

-, .IV[B] -+ A -+ C.

By (B3) we obtain

-,.IV[B] -+ A -+ .IV[B].

By Lemma 13 and (B4) we obtain


-, -,A -+ .IV[B].

By Lemma 10 the formula A -+ -, -, A is deducible so by Lemma I we obtain A -+ .IV[B].


Theorem 18.3. r~[A, A].


r A -+ ~[A, A] and rIA -+ ~[A, A].

By Lemma 17 we obtain r ~[A, A].

Theorem 18.4. r%[O' =0].

Proof As an (A I)-axiom we have

r,O'=O.

By Lemma 19 we have

rIO' =0-+ %[0' =0].


r%[O'=O].

Theorem 18.5. %[,A],%[B] r %[A -+ B].

Proof By Lemma 20 from %[,A] and %[B] we obtain

IA-+ %[A-+ B] and B-+ %[A-+ B].

By Letpma 18 we obtain

(A -+ B) -+ %[A -+ B].

By Lemma 19 we have

r I (A -+ B) -+ %[A -+ B].


%[A-+ B].

Theorem 18.6. ~[F[u]] r ~[\lxF[x]], provided the free number variable u does not occur in the conclusion.

Proof By Lemma 20 from ~[F[u]] we obtain

1~[\lxF[x]] -+ F[u].

146

By (B7) we obtain

-,&'[Vxff[x]] - Vxff[x].


-'Vxff[x] - &,[Vxff[x]].

By Lemma 19 we have

~Vxff[x] - &,[Vxff[x]].


&,[Vxff[x]].

Theorem IS.7. Vxff[x] - %[ff[t]] ~ %[Vxff[x]].

Proof By Lemma 20 from the hypothesis

Vxff[x] - %[ff[t]]

we obtain

ff[t] - Vxff[x] - %[Vxff[x]].

Using the (A2)-axiom

Vxff[x] - ff[t]

by Lemma 1 we obtain

Vxff[x] - Vxff[x] - %[Vxff[x]].

By (B4) we obtain

Vxff[x] - %[Vxff[x]].

By Lemma 19 we have

-,Vxff[x] - %[Vxff[x]].


%[Vxff[x]].

VII. Pure Number Theory

18 .. The Formal System PN for Pure Number Theory 147

Theorem 18.8 (Cut rule). &I[A], A -+ B I- &I[B].

Proof By Lemma 20 from &I[A] we obtain

-, &I[B] -+ A.

By Lemma 1 using A -+ B we obtain

-,&I[B] -+ B.


-,B-+ &I[B].

By Lemma 19 we have

I-B-+ &I[B].


&I[B].

Theorems 18.3-18.8 show that the system PN has all the deduction properties which were laid down in §3 (p. 20) for the formal system CP of classical predicate calculus. For by Theorems 18.3 and 18.4 all the axioms of the system CP are deducible in PN (with 0' = 0 for 1.) and by Theorems 18.5-18.7 the basic inferences (S 1 )-(S3) of the system CP are permissible inferences in the system PN. By Theorem 18.8 cuts, which were shown to be permissible inferences in the system CP, are also permissible in PN. By Theorems 18.1 and 18.2, axioms (A3)-{A8) and the inference rule (B8) of complete induction the system PN is also sufficient for all the requirements of arithmetic which lie in the domain of classical predicate calculus.

6. The Consistency of the System PN

It was G. Gentzen who in 1936 first proved the consistency ofa formal system for pure number theory which is equivalent to the system PN. We shall give two different kinds of consistency prooffor PN. The first is in §19 by an interpretation in the system FT of functionals of finite type following K. GOdel in 1958 and the second in §23.2 by an embedding in a formal system EN which will be proved consistent by cut-elimination. Like the original proof of G. Gentzen, besides elementary methods both proofs only require transfinite induction up to eo.


§ 19. Interpretation of PN in FT

1. Sequences of Terms of the System FT

In this section we use w, Wi' X, Xi' y, Yi' Z, Zi to denote finite (possibly empty) sequences of distinct variables of the system FT and we use a, ai' b, bi for finite (possibly empty) sequences of terms of the system FT.

a I-a;. -:an denotes the sequence of terms obtained by concatenating the sequences of terms a1,· •• , an'

If a is a sequence of terms arl ... Pm'l, ... , a: 1 ••• 0"",'" and b is a sequence of terms br' , ... , b::,m then a(b) denotes the sequence of terms

The sequence of terms a(b) is empty if, and only if, a is empty. If b is empty then a(b) is the same as a. For brevity we write ab for a(b) provided the sequence of terms b is denoted by a single symbol.

If a(b 1-b2 - ••• -bk ) is defined then by the definition of sequences of terms this sequence is the same as ab 1 ••• bk •

When a:=a~\ ... ,a::,'" and b:=b~I, ... ,b~" are sequences of terms we use the following conventions:

1. a",b ("a is of equal type with b") means that m=n and t1i='i for all i= I, ... , n. (In particular this holds if a and b are empty.)

2. I-a = b means that a'" band aI; = b~; for all i = I, ... , n is deducible in FT.

I-A-+ a=b

means that a'" band

A-+a?=b?

is deducible in FT for all i= I, ... , n. (I-a =b and I-A -+ a=b are also to hold if a and b are empty.)

Given sequences of terms a : = all, ... , a;" and b : = b II , ... , b;" let

D[cO, a, b]

be the sequence of terms

Using (DO)and (D[/) (§17, p.1l9) we obtain

I-cO=O-+ D[cO, a, b]=a I-cO= [/UO -+ D[cO, a, b] =b.

19. Interpretation of PN in FT 149

Lemma 1. Given two sequences of terms a and b, then up to equality of type, there is a uniquely determined sequence of variables x such that

xa~b.

Proof If a is a sequence of terms ar l , ••• , a::'~ and b is a sequence of terms b~', ... , b~n, then xa ~ b holds if, and only if, x is a sequence of variables

Lemma 2. Given sequences Xl' ... , X k of distinct variables and a sequence of terms a there is a sequence of terms b containing no variable occurring in the sequence Xl' ... , X k , such that

Proof If x;-: .-:Xk is the sequence of variables yr l , ••• , y~~ and a is the sequence of terms a~', ... , a~n then the assertion holds for the sequence of terms

2. The Formal System QFT

We obtain the interpretation of PN in FT by using formulas in a language QFT which is a quantificational extension of the language of FT.

Inductive Definition oftheformulas ofQFT and the free occurrences of variables in such formulas.

1. Every equation aO = bO between terms aO, bO of type 0 is a formula of Q FT. A variable isfree in this formula if, and only if, it occurs in aO or bOo

2. If A and B areformulas ofQFT then so too is (A ~ B). A variable isfree in this formula if, and only if, it is free in either A or B.

3. If F is aformula of QFT and x' is a variable not bound by a quantifier 'ix' or 3x' then 'ix'F and 3x'F are also formulas of QFT. A variable isfree in such a formula if, and only if, it is different from x' and is free in F.

A formula ofQFT is said to be quantifier-free if it contains no 'i or 3 quantifier. The quantifier-free formulas of QFT are exactly the basic formulas of the system FT (see p. 129).

We use the usual arrangements for omitting brackets and the notation r ~ B, where r denotes a sequence offormulas, for the formulas ofQFT as we did for FT.

A sequence of terms a is said to be free for a formula F of QFT if no variable y' occurring in a occurs in a quantifier 'iy' or 3y' in the formula F.

We abbreviate the formulas

ff[a~', . .. , a~n]


ofQFT to

Vx~[x], 3x~[x], ~[a],

where x denotes the sequence of variables x~., ... , x~", a the sequence of terms a~., ... , a~" and ~ an n-place nominal form. We allow the case n=O when x and a are empty. The notation ~[a] already assumes that the sequence of terms a is free for the formula ~[x].

Definition of negation in QFT:

Let ,A be the formula A ---> 9'0=0.

It is not necessary to give a deduction procedure for formulas of QFT in order to give the interpretation ofPN in FT. However, we shall give a deduction procedure for QFT since it will motivate the choice of the interpreting formulas.

Axioms of the system QFT: 1. All basic formulas which are deducible in the system FT. 2. The V-axioms

Vx~[x] ---> ~[a]

and 3-axioms

~[a] ---> 3x~[x]

for arbitrary sequences of variables x and sequences of terms a ~ x. 3. The IP~-axioms (Independence of premise)

(Vxd[x] ---> 3y g6[y]) ---> 3y(Vxd[x] ---> 2[y])

for quantifier-free formulas d[x]. 4. The M'-axioms (Markov's principle)

,,3xd[x] ---> 3xd[x]

for quantifier-free formulas d[x]. 5. The AC-axioms (Axioms of choice)

Vx3y~[x, y] ---> 3zVx~[x, zx]

for the sequences of variables x, y, z such that y~ zx.

Basic Inferences of the system QFT 1. The basic inferences (Str), (Cut) and (CI) of the system FT for formulas of

QFT (see p. 114).

19. Interpretation ofPN in FT 151

2. The V-inferences

r ~ ff[x] f- r ~ Vxff[x]

provided that the variables in the sequence x do not occur in ff and do not occur free in r.

3. The 3-inferences

ff[x] ~ B f- 3xff[x] ~ B

provided that the variables in the sequence x do not occur in ff and do not occur free in B.

Corollary. Every formula

A~A

is deducible in QFT.

Proof A ~ A is a special case of an V-axiom or 3-axiom with an empty sequence of variables.

Remark. The formal system QFT corresponds to a fragment of the formal system obtained by adding the axioms IP~, M' and AC to Heyting arithmetic HAW. (See A.S. Troelstra: Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics 344. Springer-Verlag 1973, p. 46 and 238.) QFT lacks the connectives A and v of HAw.

Lemma 3. The following are deducible in QFT: (1) (A~ B)~ (B~ C)~ A~ C (2) (A~B)~-'B~-,A.

Proof The formulas

(A~ B)~ A~ B and B~ (B~ C)~ C

are deducible. Hence by a cut

(A~ B)~ A~ (B~ C)~ C.

Bya structural inference we have (1) (A~ B)~ (B~ C)~ A~ C (2) is a special case of (1) with 9'0 = 0 for C.

Lemma 4. The folloWing are permissible inferences of the system QFT: (1) A~ Bf- (B~ C)~ A~ C


(2) A----+BI-,B----+,A (3) ·B ----+ C I- (A ----+ B) ----+ A ----+ C.

Proof This follows from Lemma 3 by cuts and a structural inference.

Lemma 5. Thefollowing are deducible in QFT: (1) 9'0=0----+ B (2) ,A----+A----+B.

Proof of (1) by induction on the length of the formula B. 1. Suppose B is an equation aO = bOo Then the assertion holds by Theorem 17.6. 2. Suppose B is a formula A ----+ C. Then by I.H.

is deducible. The assertion follows by a structural inference. 3. Suppose B is a formula Vxff[x] or 3xff[x] where the nominal form ff

contains no variable in the sequence x. Then by I.H.

9'0=0----+ ff[x]

is deducible. The assertion follows by an V-inference or cut using an 3-axiom. (2) follows from (1) by Lemma 4(3).

Lemma 6. If A is a quantifier-free formula, then

A----+B, ,A----+BI-B

is a permissible inference of the system QFT.

Proof By Theorems 17.5 and 17.6 the formulas

X[A] =0----+ A and X[A]=9'x°----+ ,A

are deducible in FT and therefore also in QFT. Using cuts with

A----+ Band ,A----+ B

we obtain

X[A]=O----+B and X[A]=9'x°----+B.

Using a structural inference and a basic inference (CI) we obtain

X[A] = X[A] ----+ B.

Hence we obtain B.


Lemma 7. If A is a quantifier-free formula then the formula

is deducible in Q Ff.

Proof The formulas

A -+ -, -,A -+ A and -,A -+ -, -,A -+ A

are deducible in QFT. Hence by Lemma 6 we obtain -, -,A -+ A. We say that two formulas Fand G are equivalent in QFT if the formulas F -+ G

and G-+ Fare deducible in QFT.

Theorem 19.i. Let Xl' Yl' Zl' X 2 , Y2' Z2 be sequences of distinct variables which do not occur in the nominal forms d and f!4 and which are such that Y 1 ~ Z 1 X 1 Y 2 and X2 ~ Z 2X 1. Let d[x 1, Y 1] and .?l[x 2' Y 2] be quantifier-free formulas, then the formulas

and

are equivalent in QFT.

Proof 1. The formula

is deducible in QFT by Lemma 5(2). Using a cut with an 3-axiom we obtain

By Lemma 4(2) and a cut using the formula

(which is deducible by Lemma 2) we obtain

By an \i-inference and Lemma 4(2) we obtain


Using a cut with an M'-axiom we obtain

By Lemma 3(2) the formula

is deducible and hence by a cut and a structural inference we obtain

Obviously the formula

is also deducible in QFT. Hence by Lemma 6 we obtain

By Lemma 4(3) from an V-axiom we obtain

By a cut and an V-inference we obtain

Bya cut using an AC-axiom we obtain

By a cut using an 3-axiom and an 3-inference we obtain

Bya cut using an IP~-axiom we obtain

From an 3-axiom by Lemma 4(1) we obtain

By a cut and an V-inference we obtain

(3Xl VY1'-91[x 1 , Yl] - 3xz VYz~[xz' Yz]) - Vx 13x i"'zVYz(.·91[x 1 , zYz]- ~[xz' Yz])·

19. Interpretation of PN in FT

By a cut using an AC-axiom we obtain

and

(3xl VYl'-G1[x l , Yl] ~ 3x2 VY2.?4[X2' Y2]) ~ 3Z;-Zl Vx;h(d[x l , Zl Xl Y2] ~ .?4[Z2Xl' Y2])·

2. By structural inferences and a cut from the V-axioms

we obtain

By an V-inference and a cut using an 3-axiom we obtain

By structural inferences and 3-inferences we obtain

3Z2~Zl Vx{"h(d[x l , Z 1 Xl Y2] ~ .?4[Z2Xl' Y2]) ~ 3x 1 VYl d[Xl' Yl] ~ 3Xi VY2.?4[X2, Y2].

This completes the proof of Theorem 19.1.

3. Interpreting Formulas

We use formulas

3xVyd[x,y]

155

of the system QFT as interpreting formulas (for formulas of the system PN) where x, yare sequences of distinct variables which do not occur in the nominal form d and d[x, y] is a quantifier-free formula. x and Y may be empty. The choice of bound variables in the interpreting formulas is not significant. That is to say: two interpreting formulas

where Xl'" x2 and Y 1 '" Y2 are identified with each other. We regard every free number variable as a variable of type 0 and every number term t' as an abbreviation for9't. The number terms of the system PN are then to be understood as terms of


type 0 in QFT where + and· are to be understood according to the definition in §17 (p. 120).

Inductive Definition of an interpreting formula F* for each formula F of the system PN.

1. If F is an equation s = t, then F* is the same equation where s, t are understood as terms of type o.

2. If F is a formula A ---+ B and A *, B* are the interpreting formulas

where the variables in the sequences Xl' Y l' xz, Yz are chosen to be distinct from one another and not to occur in the nominal forms .91, f!l then F* is the interpreting formula

wherezl, Zz are chosen by Lemma I so thatYl"" ZlX1YZ andxz '" ZZxl . The variables in the sequences Zl' Zz are to be distinct from each other and from the variables in the sequences Xl' Yz and are not to occur in the nominal forms .91, f!l. (This choice of F* is motivated by Theorem 19.1.)

3. If Fis a formula Vxff[x] where the bound number variable X does not occur in the nominal form ff, u is a free number variable which also does not occur in ff and ff[u]* is the interpreting formula

where u is distinct from the variables in the sequences Xl' Yl and does not occur in the nominal form .91, then F* is the interpreting formula

where Z is chosen by Lemma 1 so that Xl '" ZU. The variables in the sequence Z are to be distinct and also distinct from u and the variables in the sequence Y 1 and not to occur in the nominal form.9l. UJl denotes the sequence of variables obtained by concatenating the variable u (of type 0) with the sequence Y l' (This choice of F* is motivated by an AC-axiom.)

Definition. An interpreting formula

3xVy.9I[x,y]

is said to be valid if there is a sequence of terms a, in which no variable in the sequence Y occurs, such that a'" X holds and the quantifier-free formula .9I[a, y] is deducible in the system FT.


Theorem 19.2 (Interpretation Theorem). If F is a formula deducible in the system PN then its interpreting formula F* is valid.

Corollary. The formal system PN is sententially consistent. That is to say: There is no formula A such that A and ....,A are both deducible in PN.

Proof ....,A is defined to be the formula A --> 0' =0. If A and A --> 0' =0 were deducible in PN then by § 18 (Lemma 2) the formula 0' = 0 would also be deducible in PN. But by Theorem 17.2 its interpreting formula YO = 0 is not deducible in FT and therefore is not valid. It follows by Theorem 19.2 that 0' = 0 is not deducible inPN.

We prove Theorem 19.2 by induction on the deduction, viz. by induction on the length of an assumed deduction of F in PN. We write I-A to indicate that A is deducible in the system FT.

4. Interpretations of the Axioms of the System PN

We first prove that the interpreting formula F* of each axiom F of the system PN is valid.

1. Suppose F is an axiom

(AI) A --> A.

Then F* is an interpreting formula

By Lemma 2 there are sequences of terms aI' a2 in which no variable from the sequences x I' Y2 occurs such that

Hence we have

Thus F* is valid. 2. Suppose F is an axiom

(A2) Vx%Tx] --> %[t].

Then the formulas Vx%[x] and %[t] have interpreting formulas


Hence F* is an interpreting formula

By Lemma 2 there are sequences of terms ao' a l , a2 in which no variable from the sequences Xl' Y2 occurs such that

Thus F* is valid.

3. Suppose F is an axiom

(A3) u' =O~ A.


3xVy(9"u=0~ d[x,yJ).

By Theorem 17.6 we have

f-9"u=O~ d[x,y].

Thus F* is valid. 4. Suppose F is an axiom

(A4) u'=v'~u=v.

Then the interpreting formula F* is the quantifier-free formula

which is deducible in FT by Theorem 17.9 and therefore valid. 5. Suppose Fis one of the axioms (A5)-(A8). Then F* is an equation which is

deducible in FT by § 17 (p. 120) and therefore valid.

5. Interpretations of the Basic Inferences in the System PN

We now prove that assuming the interpreting formulas of all premises of a basic inference of the system PN are valid, then the interpreting formula of the conclusion is valid.


I. Suppose G is the conclusion of a basic inference

(BI) B~A-+B.

Then B* is an interpreting formula

and G * is an interpreting formula

By hypothesis there is a sequence of terms a2 in which no variable from the sequence Y2 occurs such that

Then by Lemma 2 there is a sequence of terms b2 in which no variable from the sequences Xl' Y2 occurs such that

It follows that

Thus G* is valid. 2. Suppose F is the premise and G the conclusion of a basic inference


{ 3Z4~Z3~Z;Z I VXI~X2~X3Y 4(d[x l , Z I Xi X 2 X 3Y 4] -+ P4[X 2 , Z2 X I X 2X 3Y 4]

-+ CC[X3 ,Z3 X I X 2X 3Y4] -+ ~[Z4XI X 2 X 3' Y4])

and G* is an interpreting formula

{3W4~W2"'W3~WI VXI~XI~X3~X2Y4(d[XI' W I X I X 3X 2Y4] -+ CC[x 3 , W 3X I X 3X 2Y4]

-+ P4[Xz, W2X I X 3X 2Y4]-+ ~[W4XIX3X2'Y4])'

By hypothesis there are sequences of terms aI' a 2 , a 3 , a 4 in which no variable from the sequences Xl' X 2 , X 3' Y4 occurs such that

~{d[XI' a l x l x 2x 3Y4] -+ ~[X2' a 2x l x 2x 3Y4]

-+ CC[x 3 , a 3x I X 2 X 3Y4] -+ ~[a4xI X 2X 3' Y4].


By Lemma 2 there are sequences of terms bl , b2, b3, b4 in which no variable from the sequences Xl' X 2 , X 3 , Y 4 occurs such that

and

Using the equality rules and a structural inference we obtain

f-{d[X I, bl x 1X3X2Y4] -> 'i&'[X3' b3x IX3X2Y4]

-> 86[x2, b2x IX3X2Y4] -> E&[b4x I X3X2' Y4].

Thus G* is valid. 3. Suppose F I , F2 are the premises and G the conclusion of a basic inference

Then F{, Fi are interpreting formulas

3Z3~Z2~Z I VXI~Z;Y3(d[XI' ZIXI X2Y3] -> 86[X2' Z2 XI X2Y3] -> 'i&'[Z3 XIX2' Y3])' 3Z4~Z5 VX3~Y 4('i&'[X3, Z 5X3Y4] -> E&[Z4X3' Y 4])'


3W4~W2~WI VXI~X2~Y4(d[XI' WIXIX2Y4] -> gH[X2' W2XIX2Y4] -> E&[W4XIX2'Y4])·

By hypothesis there are sequences of terms, al , ••• , a5 in which no variable from the sequences Xl' X2, X3, Y3' Y4 occurs such that

and

and

f-{d[X I, a l Xl x 2(a 5(a 3x I X2)Y 4) -> 86[X2' a2x I x 2(a 5(a 3XI X2)Y 4)]

-> 'i&'[a3xIX2' a5(a3x l x 2)Y4]

19. Interpretation ofPN in FT

By a cut we obtain

l-{d[X I, a l Xl x 2(a 5(a 3x I X 2)Y 4)] -4 .?4[X2' a2x I x 2(a 5(a3x I X 2)Y 4)]

-4 f0[a4 (a3 x l x 2), Y4].

161

By Lemma 2 there are sequences of terms bl , b2 , b4 in which no variable from the sequences Xl' X 2 ' Y4 occurs such that

I-bl Xl X 2Y 4 =al Xl x 2(a 5(a3x I X 2)Y 4)'

I-b2x l X 2Y 4 = a2x I x 2(a 5(a 3x I X 2)Y 4)'

I-b4 x I X 2 =a4 (a3x l x 2 )·

Hence we have

I-d[xl , bl x l X 2Y4] -4 .?4[X2' b2x I X 2Y4] -4 f0[b4x I X 2'Y4].

Thus G* is valid. 4. Suppose F is the premise and G the conclusion of a basic inference

(B4) A -4 A -4 B I- A -4 B.


and G * is an interpreting formula

By hypothesis there are sequences of terms ao, aI' a2 in which no variable from the sequences X O' Xl' Y2 occurs such that

Substituting Xl for Xo by Theorem 17.1 we obtain

By Lemma 2 there are sequences of terms bl , b2 in which no variable from the sequences X I' Y 2 occurs such that

(2) I-b2x I =a2 x l x l

I-b l x I Y2 =D[X [d[XI' aox l x I Y2]]' al x l x l Y2' aox l x l Y2].

Hence we have (see p. 119)

(3) I-X[d[x l , aoX l x l Y2]] =0-4 bl x I Y2 =al x l x l Y2

(4) I-X[d[xl , aox l x I Y2]] =//uo -4 bl x I Y2 =aOx l x 1Y2].



By a cut with (1) we obtain

Using (2) and (3) we obtain


Using (4) we obtain

By the Corollary to Theorem 17.5 from (5) and (6) we obtain

Thus G* is valid. 5. and 6. Suppose F is the premise and G the conclusion of a basic inference

(B5) u=u- A I- A

or

(B6) 37'[s] I- s=t- 37'[t].

Then trivially F* is valid implies G* is valid. 7. Suppose F is the premise and G the conclusion of a basic inference

(B7) A - 37'[u] I- A _ 'v'x37'[x].




By hypothesis there are sequences of terms aI' a2 in which no variable from the sequences XI' Y 2 occurs such that

By Lemma 2 there are sequences of terms b1 , b2 in which no variable from the sequence Xl~U~Y2 occurs such that

Hence

Thus G* is valid. S. Suppose F l , F2 are the premises and G the conclusion of a basic inference

(BS) $'[OJ, F[uJ -4 $'[u'J I- $'[t].

Then Fi, Fi are interpreting formulas

:3xVyd[x, y, 0], • :3z2~Zl Vry(d[x, ZIXY, u] -4 d[Z2X, y, 9'u]),

where u does not occur in the nominal form d and G* is an interpreting formula

:3xVyd[x, y, t].

By hypothesis there are sequences of terms aD, aI' a2 in which no variable from the sequences x, y occurs such that

(1) I-d[ao, y, 0] I- d[x, a 1xy,u] -4 d[a2x, y, 9'u].

We may assume that u does not occur in aD. By Lemma 2 there are sequences of terms b1 , b2 in which neither u nor any variable from the sequences x, y occurs such that

I-biu=ai (i= 1, 2).

Hence

(2) I-d[x, b1 uxy, u] -4 d[b 2ux, y, 9'u].

By Theorem 17.12 there is a sequence oftermsJin which neither u nor any variable from the sequences x, y occurs such that


(3) I-fO=ao (4) 1-j{!7u)=b2u(ju).

From (1) and (3) we obtain

(5) 1-.91[10, y, 0].

From (2) and (4) substitutingfu for x we obtain

(6) I-d[fu, bi u(ju)y, u] ~ d[j{Y'u), y, !7u].

From (5) and (6) by Theorem 17.13 we obtain

I-dEft, y, t].

Thus G * is valid. This completes the proof of Theorem 19.2.

Part C

Subsystems of Analysis

Chapter VIII

Predicative Analysis

In this chapter we consider formal and semi-formal systems in which the Peano axioms hold for number variables and there is quantification over predicate variables. In these systems the real numbers are definable by predicators as certain sets of rational numbers and universal and existential statements about real numbers are formalizable. Here certain subsystems of classical analysis will be delimited. We restrict ourselves in this chapter to predicative subsystems of analysis, that is, to systems which are interpreted in such a way that a concept which is defined under the assumption of a collection of concepts does not itself belong to this collection.

§20. Systems of L1 ~-Analysis

The systems in this section have the same language as for classical analysis but are restricted in the definition of predicators (sets) by Ll ~-comprehensioo for which a predicative interpretation will be made possible (in §22).

1. The Formal Language of Second Order Arithmetic

As primitive symbols we use 1. Denumerably infinitely many free and bound number variables and predi

cate variables. (All predic::tte variables are to be I-place.) 2. The symbols 0, I, 1.., --., V and A.. 3. Symbols for n-place calculable arithmetic functions and n-place decidable

arithmetic predicates (n;?l: 1). (The conditions governing the use of these symbols will be given on page 168.)

4. Round brackets and comma. We use nominal forms in the usual way and denote them by capital script

letters.

Inductive definition of terms: 1. The symbol 0 is a term. 2. Every free number variable is a term.

168 VIII. Predicative Analysis

3. If t is a term then so too is t . 4. If f is a symbol for an n-place calculable arithmetic function (n~ 1) and

t l' ... , tn are terms, then f(t l' ... , tn) is also a term. Terms built up according to 1. and 3. only are called numerals. A term is said

to be numerical if it contains no free number variables. The prime formulae are: I. The symbol .1.., 2. ~(t1"'" tn) where ~ is a symbol for an n-place decidable arithmetic

predicate (n~ 1) and t1 , • •• , tn are terms. A prime formula is said to be constant if it contains no free number variable.

Inductive definition of formulas and predicators: 1. Every prime formula is a formula. 2. Every free predicate variable is a predicator. 3. If P is a predicator and t a term, then P(t) is a formula. 4. If A and B are formulas then so too is (A ---+ B). 5. If ff[O] is a formula and x is a bound number variable which does not

occur in ff, then Itxff[x] is a formula and hff[x] is a predicator. 6. If U is a free predicate variable, ff[ U] is a formula and X is a bound

predicate variable which does not occur in ff then ItXff[X] is a formula. A formula is said to be simple if it is not of the form (A ---+ B). A predicator is

said to be elementary if it contains no bound predicate variables. By the length of a formula we mean the number of symbols ---+, It and A. which occur in the formula.

Syntactic symbols: a, b, c, d for free number variables, x, y, z for bound number variables, U, V, W for free predicate variables, X, Y, Z for bound predicate variables, s, t for terms, P, Q for predicators, A, B, C, D, E, F, G for formulas, i,j, k, m, n for natural numbers and their corresponding numerals. (A natural number n corresponds to the numeral in which the symbol occurs precisely n times.) We shall also use these syntactic symbols with indices.

Conditions for the use of arithmetic functions and predicates symbols: 1. A symbol f for an n-place calculable arithmetic function (n ~ 1) may only

be used as a primitive symbol provided there is a general procedure by which a numeral may be calculated as the value of f(m 1, ... , m n) for each n numerals m1,·· .,mn ·

2. A symbol ~ for an n-place decidable arithmetic predicate (n~ 1) may only be used as a primitive symbol provided there is a general procedure which decides, for each n numerals m 1, ... , mn , whether ~(m1" .. , mn) is true or false.

20. Systems of Lll-Analysis 169

Inductive definition of the value of a numerical term: 1. The term 0 has value O. 2. If t ~s a numerical term of value m, then t' has value m'. 3. If f is a symbol for an n-place calculable arithmetic function (n~ 1) and

t l' ... , t. are numerical terms with values m l' ... , m., then f(t l' ... , t.) has the value given for f(ml' ... , m.) by the procedure for f which we required to exist.

By this definition every numerical term has a calculable value which is a numeral.

Definition of the truth-value of a constant prime formula. 1. The formula ..1 isfalse. 2. If ~ is a symbol for an n-place decidable arithmetic predicate (n~ 1) and

t l' ... , t. are numerical terms with values m l' ... , m. then ~(t 1, ... , t.) is true or false according as ~(ml' ... , m.) is decided to be true or false by the procedure which we required to exist for ~.

By this definition every constant prime formula is, decidably, either true or false.

Two formulas are said to be equivalent if they are formulas ff [s l' ... , sn] and ff[rl' ... , tn] where Si and ti(i= 1, ... , n) are numerical terms of equal values.

Theorem 20.1. Equivalence offormulae is an equivalence relation.

Proof By definition equivalence is reflexive and symmetric. A numerical term t

is said to be a maximal-numerical component of a formula F, if t occurs in F but not as a component of a longer numerical term. Bya substitution which transforms F into an equivalent formula every maximal-numerical component of F is transformed into a numerical term of equal value while every other component of F remains unaltered. Hence it follows that equivalence offormulas is also transitive.

We define positive and negative parts of formulas, P-forms, N-forms and NP-forms in the usual way (see p. 20). As before we define:

The minimal positive parts of a formula F are the simple positive parts of F. The minimal negative parts of a formula F are those negative parts of F which are not of the form (A -+ ..i). The minimal parts of F are the minimal positive and negative parts of F.

F~ G (G follows structurally from F) denotes that to every minimal positive (negative) part of F there is an identical positive (negative) part of G.

We use [7J> for P-forms, JV for N-forms, f2 for NP-forms and Iff for P-forms and N-forms as syntactic symbols.

As in §4.4 we define -,A: =(A -+ ..i)

(AAB):=-,(A-+ -,B) (A v B): =(-,A -+ B)

(A +-+ B): = -,((A -+ B) -+ -,(B -+ A» 3xff[x]: = -,Vx-,ff[x]

3Xff[X]: = -,VX-,ff[X].


For brevity we omit the outer round brackets in formulas. We also write Al ~ ... ~ An~ B for (AI ~ ( ... ~ (An~ B) .. . ». 2. The Formal System DA Axioms: (Axl) gl/[A], if A is a true constant prime formula, (Ax2) JV[A], if A is a false constant prime formula. (Ax3) .E![A, B) if A, B are equivalent formulas of length O. (Ax4) ff[aI , ... , an] (n~ 1), if for every n numerals mI , ... , mn ff[mI' " ., mn] is one of the axioms (Axl)-(Ax3).

The minimal parts indicated in axioms (Axl)-(Ax3) are called the principal parts of these axioms.

Remark. It -is not in general decidable whether a formula is an (Ax4) axiom. A formula, therefore, may only be used as an (Ax4) axiom if there is a general procedure which shows that the formula does satisfy the conditions for such an axiom.

Basic inferences: (SI) JV[,A], JV[B] I- JV[(A ~ B)], if B is not the formula . .1. (S2.0) gl/[ff[a]] I- gl/['v'xff[x]], if a does not occur in the conclusion. (S2.1) gl/[ff[U]] I- gl/['v'Xff[X]], if U does not occur in the conclusion. (S3.0) ff[t] ~ JV['v'xff[x]] I- JV['v'xff[x]] (S3.E) ff[P] ~ JV['v'Xff[X]] I- JV['v'Xff[X]], if P is an elementary predicator. (S3 M {'v'X('v'Yd[X, YJ ~ .... dY8l[x, Y]),

. ff[h'v'Yd[x, YJ] ~ JV['v'Xff[X]] I- JV['v'Xff[X]], if no bound predicate variables occur in d and 81. (S4) 8[ff[t]] I- 8[2xff[x](t)] (cut) gl/[A] , A ~ B I- C, if gl/[B] f!. C holds. (ind) ff[O], ff[a] ~ ff[a'] I- ff[t], if a does not occur in ff. (Complete induction inference)

Remark on (S3.E) and (S3.~). If ff[U] is a formula containing no bound predicate variable then 'v'Yff[y] is said to be a II~-formula and 3Yff[Y] is said to be a L~-formula. If the formula

'v'Yd[Y] +-+ 3Y8I[Y]

is deducible for a II~-formula 'v'Y dEY] and a L~-formula 3 Y86[y] then 'v'Y dEY] and 3 Y 81[Y] are said to beA ~-formulas. Similarly we call a predicator h'v'Y d[x, YJ a A ~ -predicator if there is a deducible formula

'v'x('v'Yd[x, Y]+-+3Y86[x, YJ)

where no bound predicate variables occur in d and 81. So by (S3.E) and (S3.M basic inferences of the form (S3.1) ff[P] ~ JV['v'Xff[X]] I- JV['v'Xff[X]]

20. Systems of Lll-AnaJysis 171

in the system DA are restricted to those where P is an elementary predicator or a L1~-predicator. This formation of such predicators is called L1~-comprehension. If one allows (S3.1) as a basic inference for every predicator P then with the axioms and the remaining basic inferences of the system DA one obtains full classical arithmetic of second order which is not, however, predicatively interpretable. The restriction in (S3.1) to (S3.E) and (S3.L\) which makes a predicative interpretation possible (in §22) means that the domain of the quantifiers 'if X is restricted to those predicators which satisfy ,,1 ~ -comprehension.

Inductive definition of D A f!!!' n F: 1. If F is an axiom of the system DA, then DA~' 0 F holds. 2. If DA f!!!.n Fl holds for the premise Fl of a basic inference (S2.0), (S2.1),

(S3.0), (S3.E) or (S4) of the system DA, then DA f!!!.n+ 1 Fholds for the conclusion F of that basic inference.

3. If DAf!!!i.ni Fi (i= 1, 2) holds for the premises Fl and F2 of a basic inference (SI), or (cut) of the system DA, then DA f!!!.n Fholds, where m: = max (ml' m 2) and n: = max (n 1, n2 ) + 1, for the conclusion F of that basic inference.

4. If DAf!!!i.ni Fi (i= 1, 2) holds for the premises Fl and F2 of a basic inference (S3.L\) or (ind), then DA f!!!'u Fholds, where m: = max (ml' m 2 )+ 1, for the conclusion F of that basic inference.

A formula F is said to be deducible in the system DA if there exist n&tural Plumbers m, n such that DA f!!!.n F holds. If DA f.2..n F holds then F is deducible in DA without use of the inference rule (S3.L1) and (ind). We also abbreviate DA f!!!.n Fto I- F.

3. Deducible Formulas and Permissible Inference of the System DA

Let = be the symbol for the 2-place decidable arithmetic predicate such that =(ml' m 2) is true if, and only if, the numerals m 1, 11.12 are identical. We use = as a primitive symbol of our formal language and write s = t for = (s, t).

Theorem 20.2. I- t = t.

Proof t=t is an (Axl) if t is numerical, otherwise an (Ax4).

Theorem 20.3. I- s=t~ 2 [ff[s] , ff[t]].

Proof by induction on the length of ff[s]. 1. Let ff[s] have length O. Then the formula

s=t~ 2[ff[s], ff[t]]

is an (Ax2) or an (Ax3) if sand t are numerical, otherwise it is an (Ax4).


2. Let F[s] have length >0. Then the assertion follows from the I.H. using basic inferences (SI), (S2.0), (S3.0), (S2.1), (S3.E) and (S4) as in the proof of Theorem 4.1.

By Theorems 20.2 and 20.3 aU the laws of an identity relation hold for s=t.

Theorem 20.4. 1-.2[ C, C).

Proof By Theorems 20.2 and 20.3 we have I- t=t and I- t=t-. .2[C, C). Bya cut we obtain I- .2[C, C).

Corollary to Theorem 20.4. The formal system DA is s-complete, that is, every s-valid formula of our formal language is deducible in DA. (This follows just like the Corollary to Theorem 4.1.)

By virtue of this corollary it follows in particular that all the inferences- given in Theorem 4.8 for A and v are also permissible in the system DA and also

.9[(A -. B)], .9[(B -. A) I- .9[(A +-+ B)],

.9[(A +-+ B] I- .9[(A -. B)] and .9[(A +-+ B)] I- .9[(B -. A)].

Theorem 20.S (Structural inference rule). If FI!. G holds, then F I- Gis apermissible inference.

Proof If F I!. G holds, then

F-.F,FI-G

is a cut. Using I- F -. F (by Theorem 20.4) the assertion follows.

Theorem 20.6 (Basic properties of quantifiers).

a) I- VxF[x] -. F[t] and I- F[t] -. 3xF[x] b) I- VXF[X] -. F[P] and I- F[P] -. 3XF[X], if P is an elementary or

Af-predicator.

Proof By Theorem 20.4,

I- F[t] -. VxF[x] -. F[t] and I- ,F[t] -. F[t] -. ,Vx,F[x]

hold. Assertion a) follows using basic inferences (S3.0). Assertion, b) follows similarly using basic inferences (S3.E) and (S3.A).

Theorem 20.7 (Substitution rules). Thefollowing are permissible inferences:

a) F[a] I- F[t], if a does not occur in F. b) F[U] I- F[P], if U does not occur in F and P is an elementary or Af

predicator.

20. Systems of L1 t-Analysis 173

Proof This follows from Theorem 20.6 using basic inferences (S2.0) and (S2.1).

Theorem 20.8 (Introduction of existential quantifiers). The following are permissible inferences:

a) %[ff[a]] f- %[3xff[x]], if a does not occur in the conclusion. b) %[ff[U]] f- %[3Xff[X]], if U does not occur in the conclusion.

Proof Bya structural rule we obtain %[-, -,ff[a]] from %[ff[a]] and then by a basic inference (S2.0) %[-,Vx-,ff[x]]. b) follows similarly using a basic inferenc;e (S2.1).

Theorem 20.9 (Extensionality Theorem).

f- Vx(d[x] +-+ &I [x]) - ff[..hd[x]] - ff[AX &I[x]].

Proofby induction on the length of the formula ff[U]. If U does not occur in ff then the assertion holds by Theorem 20.4. If ff[U] is a formula U(t), then the assertion follows by basic inferences (S4) from

f- Vx(d[x] +-+ &I [x]) - d[t] - &I[t].

In. all the remaining cases the assertIon follows from the IH.

Corollary. Vx(d[x] +-+ ~[x]), ff[AXd[x]] f- ff[AX &I[x]] is a permissible inference.

Let < be the symbol for the 2-place decidable arithmetic predicate such that «ml , m2 ) is true if, and only if, the numeral m l is shorter than the numeral m2 •

We use < as a primitive symbol of our formal language and write s < t for < (s, t).

Theorem 20.10 (Induction Theorem). If I is a symbol for a I-place calculable arithmetic function then

f- Vx(Vy(I(y)<I(x)- ff[y])- ff[x])- ff[t] holds.

Proof Let

F:=Vx(Vy(I(y)<I(x)- ff[y])- ff[x]), ~[t]: = Vx(I(x)< t - ff[x]).

Then we have

f-F- ~[O]

f- (F - ~[a]) - (F - ~[a']).

f- F - ~[I(t)'] follows by a basic inference (ind).


Hence

f- F~ $'[t]

which was to be proved.

4. The Semi-Formal System DA *

In DA * we use the same formal language of second order arithmetic as before but now exclude free number variables. Thus in DA * every term is numerical and every prime formula constant.

Axioms of the system DA*: (Axl)-(Ax3) as in DA.

Basic inferences of the system DA *: As in DA, but replacing (S2.0) and (ind) by

(S2.0*) &'[$'[n]] for every numeral n f- &'[Vx$'[x]].

All axioms and basic inferences of the system DA * are to be restricted to formulas in which no free number variable occurs. We call the system DA* semi-formal since, as opposed to formal systems, it has basic inferences (S2.0*) with infinitely many premises.

In what follows small Greek letters (possibly with iI1dices) always denote ordinal terms of the system OT (of §14).

Inductive definition of DA * ~ F and DA * ~ F: 1. If Fis an axiom of the system DA*, then DA*~Fand DA* ti Fhold for

every ordinal term IX of the system ~T. 2. If DA * f!i Fi holds where lXi < IX for every premise Fi of a basic inference of

the system DA *, then DA * ~ F holds for the conclusion F of this basic inference. 3. If DA * IJi Fi holds where lXi < IX for every premise Fi of a basic inference of

the system DA * which is not an inference (S3.d), then DA * ~ F holds for the conclusion F of this basic inference.

4. If DA*17 Fi holds where lXi+W:;:;1X (i= 1,2) for the premises Fl and F2 of an inference (S3.d) of the system DA *, then DA * ~ F holds for the conclusion F of this inference.

Corollaries. 1. If DA * ~ F (or DA * ~ F) and IX < p, then DA * fl!. F (or DA * ~ F) also holds.

2. IfDA* ~ F, then DA* ~ F.

A formula F is said to be deducible (or distinguished deducible) in DA* with order IX, if DA * ~ F (or DA * ~ F) holds.

Theorem 20.11 (Replacement rule). If DA * ~ F (or DA * ~ F) holds and F, G are equivalent formulas then DA * ~ G (or DA * ~ G) holds.

20. Systems of LI~-Analysis 175

Theorem 20.12 (Structural rule). q DA * I!. F (or DA * 17 F) and F j!. G hold for formulas F, G of the system DA*, then DA* I!. G (or DA* 17 G) holds.

Both these theorems are proved by transfinite induction on IX.

5. Embedding DA in DA *

If F is a formula of the system DA then we denote by F* a formula of the system DA * which results from F when all the free number variables occurring in Fare replaced by numerals. In the following we write ro· m + n for ro· m + roO • n.

Theorem 20.13 (Embedding Theorem).

DA p!!.n F implies DA* l7'm+ n F*.

Proofby induction on m with subsidiary induction on n. Suppose DA p!!.n Fholds. 1. m=O and n=O. Then Fis an axiom of the system DA and F* is an axiom

of the system DA *. Therefore DA * '* F* holds. 2. m#O or n#O. Then F is the conclusion of a basic inference BI of the

system DA. . 2.1. If BI is not a basic inference (S2.0) (S3.~) or (ind) then the assertion

follows immediately from the I.H. 2.2 Suppose BI is a basic inference (S2.0). Then Fis a formula &'[V'xF[x]] and

we have DA p!!.no &,[F[a]] where a does not occur in F and n=no+ 1. F* is a formula&'*['v'xF*[x]]. ByI.H. DA* ji!'m+no &'*[F*[k]] holds for every numeral k. Using a basic inference (S2.0*) we obtain DA* l7'm+n F*.

2.3 Suppose BI is a basic inference (S3.~). Then we have DA p!!,.n, F; for the premises Fi (i=I,2) of this inference where m=max(m1,m2 )+1 and n=O. F* is the conclusion of a basic inference (S3.~) of the system DA * with premises Fi* such that by I.H. DA* l7'mt +nt Fi* (i= 1,2) holds. Since ro·mi+ni+ro~ro·m, DA* l7' m F* follows.

2.4. Suppose BI is a basic inference (ind). Then we have DA p!!,.n, F[O] and DA P!!2. n2 F[a] -+ F[a'] where m=max (m 1, m 2)+ 1 and n=O, a does not occur in F, F is a formula F[t] and F* is a formula F*[t*]. Let mo: = max (m 1, m 2 )

and no: = max (n 1 , n2 ). By I.H. DA* ~'mo+no F*[O] and DA* l7' mo + no F*[k]-+ F*[k'] for every numeral k hold. Using cuts we obtain DA* l7'mo +(no+k) F*[k] and hence also DA * 17' m F*[k] for every numeral k. Since t* is a numerical term DA * 17' m F* follows by Theorem 20.11.

Corollaries. 1. For each formula F deducible in DA there exists IX < ro2 such that DA* 17 F*.

2. q F is deducible in DA without use of basic inferences (S3.~) and (ind) then there exists IX < ro such that D A * 17 F * .


6. General Properties of Deduction in the System DA *

Below we consider only formulas ofthe system DA * and we write ~ Ffor DA * ~ F. ~ .. F denotes that there exists e < ex such that DA * ~ F holds. A formula Fis said to befinitely deducible if~'" Fholds.

Theorem 20.14. The following formulas are finitely deducible. a) all s-validformulas, b) Vx.F[x] -+ .F[n] and .F[n] -+ 3x.F[x], c) VX.F[X] -+ .F[P] and .F[P] -+ 3X.F[X], if P is an elementary

predicator.

Proof This follows from the second corollary to Theorem 20.13 since the formulas concerned are deducible in DA without use of a basic inference (ind).

A formula B is said to be a logical consequence of formulas A 1 , • •• , An if B can be obtained by cuts from the formulas A l' ... , An and formulas which are finitely deducible by Theorem 20.14.

Theorem 20.15.lf ~ Ai(i= 1, ... , n)holdandBisalogicalconsequenceofA1,· •• , An then ~"+w B holds.

Proof Trivial.

Theorem 20.16. lf~ .!V[.F[U]] holds where U occurs in neither .!V nor .F then JL '!v[3X.F[X] for ex<p.

Proof By Theorem 20.12 from ~ .!V[.F[U]] we obtain ~ .!V[...., ....,.F[U]] and then using a basic inference (S2.1), JL .!V[....,VX....,.F[X]]. That is JL '!v[3X.F[X]].

Theorem 20.17. If ~Vx(VYd[x, y] +-+3Y£i[x, Y]) and JL .F[A.xVYd[x, Y]] where neither d nor £i contains bound predicate variables then fL 3X.F[X] if max (ex, P)<y.

Proof By Theorem 20.12 from the second hypothesis we obtain

Using the first hypothesis a basic inference (S3.~) yields fL VX....,.F[X] -+ 1... That is fL 3X.F[X].

7. Subsystems of DA and DA*

We denote by EN (elementary number theory) the subsystem of DA in which no bound predicate variables occur. In this system the basic inferences (S2.1), (S3.E) and (S3.~). disappear.

21. Deductions of Transfinite Induction 177

We let EA (elementary analysis) be the subsystem of DA in which the basic inference (S3.il) is not allowed.

The corresponding subsystems of DA * are denoted by EN* and EA *.

§21. Deductions of Transfinite Induction

In this section we develop deductions of formalized transfinite induction (over OT) in systems and subsystems of LI ~ -analysis.

In: is one of the formal systems EN, EA, DA and ~* is one of the semi-formal systems EN*, EA*, DA* then

~ f- F denotes that the formula F is deducible in ~,

~* f!. F(~* f..!'1X F), that the formula F is deducible in ~* with order ex (with an order <IX).

The following .obviously hold:

EN f- F = EA f- F and EA f- F = DA f- F,

EN* f!. F= EA* f!. F and EA* f!. F= DA* f!. F.

By Theorem 20.13

~ f- F = ~* f..!'",2 F* also holds.

1. Formalisation of Transfinite Induction

In § 14 we defined a bijective mapping Nr from OT onto the set N of natural numbers with inverse T (p. 97). We regard N as the set of numerals and denote by ~ the numeral Nr IX of an ordinal term ex from ~T.

Corresponding to the relations < and ::::; on OT the given mapping induces 2-place decidable arithmetic predicates -< and "' defined as follows:

-«m, n): ¢> T(m) <T(n)

.,; (m, n): ¢> T(m)::::; T(n).

We use -< and "' as primitive symbols of our formal language and write

s -< t for -< (s, t), s.,;t for .,; (s, t),

Al~··· ~An~B for (AI ~ ( ... ~ (An~ B) .. . », tl -<t2 -<t3 ~ B for tl-<t2~t2-<t3~B,

tl"'t2-<t3~B for tl "'t2 ~ t2 -<t3~ B, Vx-<t~[x] for Vx(x -< t ~ ~[x]).


The relation -< is easily shown in EN to be a linear ordering relation, that is, the following formulas are deducible in EN for arbitrary terms t l' t 2' t 3 :

,t1 -<t1

t1 -<t2~ 12 -</3~ t1 -</3

,II -</2 ~ ,II =t2 ~ 12 -<11'

The combinations ex + p, cpexp, of· n, wm(P) and wm. n of ordinal terms ex, p E OT where m, n E N correspond under -< to 2-place calculable arithmetic functions +-, 4>, w, cO and w defined as follows:

~ ._{r(m)+r(n), ifr(m)#ro and r(n)#ro, +(m,n).- r 'f ( ) r () r o , 1 r m = 0 or r n = o'

A ._{cp(r(m»(r(n», ifr(m)#ro and r(n)#ro, cp(m,n).- -r 'f ( ) r () r

0' 1 r m = 0 or r n = o.

~ ._{~t(m).n, i~r(~#ro and n#O, w(m,n).- 0, Ifn-O,

ro, if r(m)=ro and n#O.

_ ._{wm(r(n», if r(n)#ro, w(m,n).- -r 'f ( )-r

0' 1 r n - o.

w(m,n):= wm·n.

We use +-, 4>, w, cO, was primitive symbols of our formal language and write

S+I for +-(s, I), wSI for w(s, I),

cOs(t) for cO(s, I), wSI for w(s, I), wS for w(s,1), wS for w(s, 1),

~t for 4>(1, t).

The following equations are true constant prime formulas for ordinal terms ex, p from OT and numerals m, n

(l/-n = wrz • n, (l/i.=wa.,

4>(ii, P) = cpexp,

cOm(f3) = wm(P),

w"=w",


From this it follows that the following equations are axioms of the system EN for arbitrary terms sand t:

&'0=0, &'t'=&"t+&',

w'O=O, w"t'=w't+w',

wo(t) = t, w.,(t) = w~(t), wt = tP(O, t), ~t = tP(I, t).

Also for all the properties of the ordinal terms 0( + p, lj>O(P, of', n, wm(P) and wm'n developed in §14 the corresponding formulas are all deducible in EN, for example

We express the progressiveness 9'~[P] of a predicate P with respect to -< and transfinite induction up to t, .F[t], with respect to -< by the following formulas:

9'~[P]:=\lX2(\lXl -<X2P(X1)-+ P(x2))

.F[t]: = \lX(9'~[X] -+ \Ix -< tX(x)).

For these the following hold:

(9'~1) EN I- 9'~[P] -+ \Ix -< tP(x) -+ P(t), if P is an elementary predicator. (9'~2) EA I- 9'~[P] -+ \Ix -< tP(x) -+ P(t) for every predicator P. (.Fl) EA I- .F[t] -+ 9'~[P] -+ P(t), if P is an elementary predicator. (.F2) DA I- .F[t] -+ 9'~[P] -+ P(t), if P is an elementary or a ..1~-predicator. (.F3) EA I- .F[t] -+ s -< t -+ .F[s] (.F4) EA I- \Ix -< t.F[x] -+ .F[t].

(9'~1), (9'~2) and (.Fl}-(.F3) are easy to prove. (.F4) is obtained as follows: From (.Fl) we have

EA I- \lx-<t.F[x] -+ 9'~[U] -+ \lx-<tU(x).

Using a basic inference (S2.1), (.F4) follows.

2. Deductions in EN

We set

Y'[P, t]: = \ly(\lx -<yP(x) -+ \Ix -<y+ tP(x))

o[P]: =,1,zY'[P, &z].


Lemma 1. For every elementary predicator P EN I- gh[P] ~ &'-r[o[P]] holds.

Proof We obtain

EN I- o[P](s)~ '<Ix<b+olaP(x)~ '<Ix <b+wsa'P(x)

from the definition of o[P] and the properties of wSt. Applying a basic inference (ind) we obtain

(1) EN I- o[P](s)~ '<Ix<bP(x)~ '<Ix <b+wStP(x)

for every term t. For ordinal terms IX, /3 and y;i=O such that 1X</3+WY an ordinal term Yo < y and a numeral n can be determined such that IX < /3 + wYo. n. Hence one can define 3-place calculable arithmetic functions f and 9 for which the following hold:

EN I- a <b+wc ~ a< b.f.w !(a,b,C)g(a, b, c)

EN I- O<c~ 1(a, b, c) <c.

By (1) with s:=f(a, b, c) and t:=g(a, b, c) we obtain

(2) EN I- O<c~ '<Ix <co[P](x)~ '<Ix <bP(x) ~ a <b+wc ~ Pea).

Further

holds. Using (&'-rl) we obtain

(3) EN I- c=O~ &'-r[P] ~ '<Ix <bP(x)~ a <b+wc~ Pea).


EN I- &'-r[P]~ '<Ix<c o[P](x)~ '<Ix<bP(x)~ '<Ix <b+wCP(x).

Using basic inferences (S2.0) and (S4) we obtain

EN I- &'-r[P] ~ '<Ix < c o[P](x) ~ o[P](c).

The assertion follows from this by a basic inference (S2.0).

Lemma 2. For every elementary predicator P and every numeral n EN I- &'-r[P] ~ '<Ix <wn(O)P(x) holds.


Pr60jby induction on n. If n~O the assertion is trivial for Wo(O) =0. We now go from n to n'~ Like P o[P] is also an elementary predicator. Hence by hypothesis EN f- &>i[o[P]] -4 Vx -<wn(O)o[P](x) hoids. Using Lemma 1 and (&'-d) we obtain

By the definition of o[P] EN f- o[P](t) -4 Vx -<sP(x) -4 Vx -<s-t-a/P(x) holds. For s: =Oand t: = WnCO) , EN f- Vx-<sP(x) and EN f- s-t- 6l =wn,(O) holds. Hence we have

The assertion EN f- &>i[P] -4 Vx -<wn,(O)P(x) for n' follows from (1) and (2).

Theorem 21.1. For every elementary predicator P and every ordinal term 0: < Go EN f- &'i[P] -4 Vx -<~P(x) holds.

Proof By Theorem 14.14 if O:<Go there exists a numeral n such that 0: < wn(O) and therefore a -< wn(O). Hence the assertion follows from Lemma 2.

3., Ded~ctions in EN*

Lemma 3. If'E is one of the formal systems EN, EA or DA and F is a formula of the corresponding semi-formal system 'E* such that 'E f- F, then 'E* pro2 F.

Proof This holds by theorem 20.13.

Lentma 4. If 'E* is one of the semi-formal systems EN*, EA * or DA * and 'E* P" ,c1>[.?[n]] where 0: #0 holds for every numeral n -</3 then 'E* ~ &>[Vx -< J3.?[x]].

Proof From the hypothesis L*P" &>[(n -< J3 -4 .?[n])] for every numeral n -< J3 holds by Theorem 20.12. For every other numeral n &>[(n-</3-4.?[n])] is an (Ax2). Hence the assertion follows by a basic inference (S2.0*).

Lemma 5. The following hold for every elementary predicator P:

a) EN* pro2 +ro." &>i[P] -4 P(m) for every numeral m -<BiZ

b) EN* f-!2 2 +w .a+3n+2 &>i[P] -4 Vx -<wnCBiZ+ I)P(x).

Proofby induction on 0:. a) If 0:=0 then a) holds by Theorem 21.1 and Lemma 3. Suppose now that

0: # O. Then for m -< BiZ there exist, by Theorem 14.14, an ordinal term ~ < 0: and a numeral n such that m -<wnCB~-t- I). Hence the assertion a) follows from the I.H. for b).

182

b) We use a subsidiary induction on n: bI) n=O. By Lemma 4 from a) we obtain

EN* f!!!2+ ro . a &'t[P] ~ '<IX-<,8a.P(X).

By Lemma 3 using (&'t 1) we obtain

Hence by Lemma 4 using a) we obtain

EN* f!!!2+ ro ' a +2 &'t[P] ~ '<Ix -<,8a.+ 1P(x).

This yields the assertion b) for n=O since WO(8a. +1) =8a. +1.

VIII. Predicative Analysis

b2) From n to n'. Like P, o[P] is also an elementary predicator. Therefore by hypothesis

EN* f!!!2+ ro ' a + 3 'n+2 &'t[o[P]] ~ '<Ix -<.wi8a.+ 1)o[p] (x) holds.

By Lemma 1, (&'tl), the definition of o[P] and Lemma 3 the following hold

EN* ~ro2 &'t[P] ~ &'t[oP]]

EN* ~ro2 &'t[o[P]] ~ '<Ix -<.wi8a.+ 1)o[p](x) ~ o[P](wi8a.+ 1))

EN* ~ro2 o[P](Wn(8a.+ I)) ~ '<Ix -<,Wn,(8a.+ I)P(x).

Using cuts the assertion

EN* f!!!2+ro'a+3n+5 &'t[P] ~ '<Ix -<,Wn,(8a. + I)P(x) holds for n'.

Theorem 21.2 EN*f!!!2+ro.a&'t[p]~'<IX-<8a.P(X) holds for every elementary predicator P.

Proof This follows from Lemmata 5a) and 4.

4. Deductions in EA and EA *

Lemma 6. EA f- '<Iy('<Ix-<.y$'[x] ~ $'[y]) ~ '<Ix -<.a[x] holds for rl<eo.

Proof As for Theorem 21.1 we have

EA f- &'t[AZ$'[Z]] ~ '<Ix -<.aAz$'[z](x)

for every predicate AZ$'[Z], since in EA there is no restriction to elementary predicators. Hence the assertion.


Lemma 7.

a) EA f- J[s] -4 J[t] -4 J[s+o/]

b) EA f- J[s] -4 J[t] -4 J[s+t]

c) EA f- J[t] -4 J[w.(t)]

d) EA f- I::/x -< tJ[8xJ -4 .jIBt] Jor t -< r o'

Proof a) The following hold by Lemma 1, (Jl) and the definition of o[U]

EA f- &.z[U]-4 &.z[o[U]]

EA f- J[t] -4 g>.z[o[U]] -4 o[U](t)

EA f- o[U](t)-4l::/x-<sU(x)-4l::/x-<s+o/U(x).

Using cuts

EA f- J[t] -4 &.z[U] -4 I::/x -<sU(x) -4 I::/x -<s+o/U(x) follows.

Further

EA f- J[s] -4 g>.z[U] -4 I::/x -<sU(x) holds.

Hence we have

EA f- J[s] -4 J[t] -4 &.z[U] -4l::/x -<s+iYU(x).

Using (S2.1) assertion a) follows. b) follows from a) by (.1'3) since s+t~s+wt. c) Trivially EA f- .1'[0] holds. Using a) for s: = 0,

By the definition of ws(t) we have

Using a basic inference (ind) we have

EA f- J[t] -4 J[ws(t)].

d) By Theorem 14.14 there are calculable arithmetic functionsJ, 9 and h such that

(1) EA f- a -< eo -4 a -< wI(a/O)

(2) EA f- a -< et -4 a -< wg(a, t)(eh(a. t) + I)

(3) EA f- 0 -< t -4 h(a, t) -< t


From EA r- J[O] and c) we have EA I- J[wf(U)(O)]. Using (1) and (.F4) we have

(4) EA I- ,IT~D].

By Theorem 21.1 EA I- J[T] holds. Using b) and c) we have

Using (2), (3) and (J4) we have

(5) EA I- 0 -< t ---+ \Ix -< tJ[~x] ----+ J[~t].

The assertion d) follows from (4) and (5).

Theorem 21.3. I/tJ.<¢lBo then EA I- J[iX].

Proo/. Letting ~[t] : = t -< r 0 ----+ J[~t] we obtain from Lemma 7d)

EA I- \ly(\lx <yff[x] ----+ ff[ yJ).

By Lemma 6 EA I- ff[PJ follows for all P<BO' Since Bp=¢IP, EA I- J[$lP] follows for all P<Bo. If tJ.<¢IBo there exists P<Bo such that tJ.<¢lp. Hence by (J3) we obtain EA I- J[iX] for tJ.<¢leo.

Theorem 21.4. EA'" /-S!~+2'''+1 J[~J holds for tJ.<ro'

Proo/by induction on tJ.. From Lemmata 7d) and 4 we have

By I.H. EA· ~w2+2'" J[~nJ holds for every numeral n -<iX. By Lemma 5 we obtain

The assertion follows from (1) and (2).

5. The Formula 9l![P, Q, t]

So far the deductions of formalized transfinite induction have essentially depended on the passage from a predicator P to the predicator o[P] which yields a step up in the transfinite induction. In order to deduce the formula J[iX] in the systems DA and DA· for higher ordinal terms Q( we need to go from a predicator P to essentially stronger predicators. To this end following a method suggested by S. Feferman [1] we use a formula 9l[P, Q, t] by which we recursively define a sequence of suitable predicators Qn(n -< t) depending on a predicator P, Hence by


using' basic inferences (S3.A) we can deduce the existence of sufficiently strong predicators.

In §14 (p. 96) a bijective mapping re of N x N onto N was defined with inverses re l and re 2 • These definitions give calculable arithmetic functions such

n(rc1 (n), re2{n)) =n,

rci(rc(n l , n2»)=ni (i= 1, 2).

We use Te, re 1• re2 as primitive symbols of our formal language and write

pes, t) for P(re(s, t)),

Ps for AyP(re(s, y)).

In addition we use hand e as symbols for the I-place calculable functions which are defined as follows:

1. If r(m) =0 or r(m)=ro then let h(m):=ij and e(m):=m. 2. If O<r(m)<ro then by Theorem 14.9 there exist unique ordinal terms

~ 1 ~ •.. ~ IY.n (n ~ 1) of the system OT such that

r(m)=w~1 + ... +w~n.

Then let

Let 9P[P, Q, t] be the formula

{VY(P(Y)A~ Q(O, y»" Vx(O..(x -< t - Vy(Q(x, y) - Vz(h(x) <z -<x-

9'[Qz' <!>(e(x) , y)]))).

lIere the restriction of the universal quantifier \;fz to t -< x is sufficient to obtain the desired deduction in DA *. The additional restrictioh of "It to hex) < z allows lis, following H. Schwichtenberg, to also obtain the desired dedu~tion in DA.

LemMa 8, OA f- Vy(Vx <y~[V, V, W,x, t] - ~[U, V, W,y, t]) where

Proof. o?l[ U, V, t] implies that· for s -< I Vs is uniquely determined by U and the Vx(x -<s). Hence the assertion follows by the Extensionality Theorem 20.9.


Let the one-place calculable arithmetic function I be defined as follows:

l. Ifr(m) =Oor r(m)=ro, let I(m):=O. 2. If 0 < rem) < r 0' then by Theorem 14.8 there are uniquely ordinal terms

a1, ... ,an,f31, ... ,f3n (n?l) of the system OT such that f3i<c/Jaif3i(i=I, ... ,n), c/Ja1f31? ... ?¢anf3n and

Then let

where L is the length of the ordinal term defined in § 14 (p. 86).

Lemma 9. DA I- .c1l[U, V, t] - 0 -<a -< t- Vx -<ag>,z[VxJ - g>,z[Val

Proof We first prove that 9'[V<, d] is implied by the following formulas (1)-(7). (l) .c1l[ U, V, t] (2) a -< t (3) Vx -<ag>,z[VxJ (4) Vy -<bVaCy) (5) Vy(l(y) < led) - y -< ¢(e(a), b) - Vz(h(a),,;;, z -<a - 9'[Vz , y]» (6) d -< ¢(e(a), b) (7) h(a),,;;, c -<a.

Now using calculable arithmetic functions we can distinguish for d the three cases 0, d1 +d2 and ¢(d1, d2) where in the last two cases l(di)<l(d). The case d=ro is excluded by (6).

l. d=O. Then trivially 9'[Vc,d] holds. 2. d=d1 +d2 where l(d) <l(d) (i= 1, 2). Then di -<¢(e(a), b) follows from (6).

Using (7) and (5) we obtain 9'[V<, dJ. Hence we obtain

and

hence 9'[ Vc, dl 3. d=¢(d1, d2) where l(dJ<I(d) (i= 1, 2). 3.l. d1 -<e(a). If c1:=c+al' then h(a),,;;,c1 -<a follows from (7). From (6)

we obtain d2-<¢(e(a),b). Using (7) and (5) we obtain 9'[Vc"d2l Hence Vx-<d2VC, (X). From (3) we obtain g>,z[Vc1l By (g>,zI) we have Vc,(d2). Since 0-< c 1 -< a using (2) and (l) by the definition of.c1l we obtain

Vz(h(c 1) ,,;;, z -< c 1 - 9'[ Vz , ¢(e(c 1), d2)])·


Since h(c1).r;; c -< c1 and e(c1) =d1 the assertion 9"[Vc, ~(dl' d2)] follows. 3.2. e(a)=d1• Then from (6) we have d2 <.b. Using (4) we obtain Vid2). By

(1), (2) and (7) from the definition of 9l we obtain 9"[ Vc ' ~(e(a), d2 )].

3.3. e(a) -<d1• Then from (6) we haved -<b. Asin3.2weobtain9"[V" ~(e(a), d)]. Using d.r;;~(e(a), d) we obtain 9"[Vc' d]. Hence we have shown

{9l[U, V, t] - a -<t- Vx -<a[11l-t[Vx ] - Vy -<bVa(y) - Vy(/(y) </(d)

DA f- - y -<~(e(a), b) - Vz(h(a).r;;z -<a - 9"[Vz , y])) - d-<~(e(a), b) - h(a).r;; c -<a - 9"[Vc ' d].

Using a basic inference (S2.0)

DA f- {9l[U, V, t] - a -< t- Vx -<a[11l-t[V J - Vy -<bViy) - d -<~(e(a), b) - Vz(h(a).r;; z -< a - 9"[ Vz , d])

follows by the Induction Theorem 20.10. With the aid of ([11I-tl) we get

DA f- [11I-t[Vc] - Vx -< ~(e(a), b)9"[V" x] - 9"[Vc , ~(e(a), b)].

From the last two formulas we have

DA f- {9l[U, V, t] - a -<t- Vx -<a[11l-t~[V J - Vy -<bViy) - Vz(h(a).r;; z -<a - 9"[Vz , ¢(e(a),b)]).

From the definition of 9l we have

The assertion follows by a basic inference (S2.0).

Lemma 10. DA f- VX3 Y 9l[X, Y, t] - oJ' -< t - .1"[&"] - .1"[~(s, 0)].

Proof By the definition of 9l we have

(1) DA f- 9l[ U, V, t] - Vy( U( y) +-+ V(O, y)).

Hence DA f- 9l[U, V, t] - [11I-t[U] - [11I-t[Vo]. Using Lemma 9 we obtain.

For the predicator P: = AX(X -< t - [11I-t[VJ) we obtain

(2) DA f- 9l[U, V, t] - [11I-t[U] - [11I-t[P].

By (.1"1) and the definition of P

(3) DA f- .1"[&"] - [11I-t[P] - &" -< t - [11I-t[VGl.] holds.


Trivially

(4) DA ~ g>z[V6js] --> V(W S , 0) holds.

Obviously h(ol) =0 and e(wS)=s. Therefor\! from the definition of9f we have

Further

(6) DA ~ Y[V(j, $(s, 0)] --> Vx -< $(s, 0) V(O, x) h~lds.

From (1)-(6) we obtain

DA ~ 9f[U, V, t] --> WS -< t --> YEwS] ---+, g>z[ V] ---+, Vx -< $(s, O)U(x).

By Theorems 20.8 and 20.6 we obtain

DA ~ V X3 Y9f[X, Y, t] --> WS -< t --> cf[wS ] --> g>z[ U] --> '<:Ix -< ¢(s, O)U(.~).

The assertiori follows by a basic inferehce (S2.1).

6. Deductions in QA

LenUna 11. For every numeral n

Proof Let ~[U, V, W, s, t] be the formula given in Lemma 8. Since run < eo it follows from Lemmata 6 and 8 that

That is

DA ~ nl(a) -<w"--> n1(a) <;(--> 9f[U, V, t] --> ~[U, W, t] --> Vy(V",(a)(Y)

- W",(a)( y)).

The assertion follows with t: = Jl:l (a) +- 1.

Lemma 12. DA ~ V X3 Y9f[X, Y, (hn] holds for every numeral n.

Praof by induction on n, Obviously DA ~ 9f[U, ~zU(nz{z)), wP] halds. So the 'lssertion holds for n = 0,

21. beductions of Transfinite Induction i89

We prove the assertiori for n' under the assumption

(l) DA f- V'X3Y9l[X, Y, w~].

r:l~mentary predicators P and Q depending on tlte free predicate variables U, V, W and the free number variable a c~n be define4 so tQat the following 1;lold:

(2.1) DA f- V'x < 6taV'y(P(x, y) +-t Vex, y» (2.2) DA /:- VxoVy(P«(hna+xo:y)-W(xo,y» (2.3) PA f- V'y(Q(y) +-+ V'z(~(wnq) ~z <wna~ Y[Vz , $(e{wna),y)]).

flere W does not occur in Q. Using the Extensionality TQeoreIl1 20.9 one obtains the fpllowing properties of P and Q from the definition of 9l: From (2.1) we have

(3.1) DA f- 9l[U, V, w,na] ~ 9l[U, p, w"al

U&ing the formula

which is deducible in DA, from (1.1)-(2,.3) we pbtain

(3.2) DA f- {9l[Q, W, wna] ~ V'y(P(wna, y) ~ . .. V'z(h(dJna) ~z <wna ~ Y[Pz , <p(e(w"a), y)]).

Frol11 (2.2) we obtain

(3.3) PA f- {9l[Q, w, w. n] ~ Vxo(.O < ~O, < wn --t V'y"(P(d/'a +xo, y)-. . V'zQ(h(xo) ~ zQ < xQ -- 5f[P cO"<!+ZQ' <p(e(xo), y)])).

If w"a ->: x < w"a' tlten (} < Xo < w" where x = cIlia + xo, h(x);= wna + h(~o) apd e(x) = e(xo). Therefore from (3.3) we have .

(3.4) DA f- {.ql[Q, w, ron] -- V'x(w"a < x ~ (bRa.' ~ Vy(P(x, y)V'z(h(x) ~ z -< X ~ Y[Pz' c!>(e(x), y)]))).

FroII1 (3.1), (3.2) and (3.4) We obtain

DA f- ~[Q, W, wn] ~ Bl(U, V, wna] ---4 9l[U, P, wna'].

By ThepreIl120,6 we obtain

DA f- £i[Q, W, c:b~] ~ ~[U, V, w~a] -d Y9l[U, Y, w"d].


Since W does not occur in Q by Theorem 20.8 we have

By Theorem 20.6 from (1) we obtain DA ~3 Y~[Q, Y, wn]. There follows

DA ~ 9f[U, V, (lla] ~ 3Y9f[U, Y, (iJna'].

By Theorems 20.8, 20.6 and using (S2.1) we successively obtain

DA ~ 3 Y9f[U, Y, (iJna] ~ 3 Y~[U, Y, (iJna,]

DA ~ VX3Y~[X, Y, wna] ~ 3Y9f[U, Y, wna']

DA ~ VX3Y~[X, Y, (iJna] ~ VX3Y~[X, Y, wna'].

As for n=O we also have DA ~ VX3 Y.~[X, Y, (iJnO]. Using a basic inference (ind) we obtain

(4.1) DA ~ VX3Y~[X, Y, (iJnt] for every term t.

There is a one-place calculable arithmetic function! such that

From (4.1) and (4.2) we obtain

With the aid of (5) we now prove that there is a iq-predicator fi[ U] such that DA ~ 9f[U, fi[U], wn']. We set

d[U, V,s,t]:=1Cl(S)<t~~[U, V,1Cl(S)-+-1]~ V(s)

.?d [U, V, s, t]: = 1C 1 (s) < t ~ (9f[ U, V, 1C 1 (s) -+- 1] 1\ V(s)).

Then

DA ~ 9f[ U, V, 1C 1 (a) -+- 1] ~ d[ U, V, a, (iJn] ~.?d[ U, V, a, (iJn'] holds.

By Theorems 20.6 and 20.8 we obtain

DA ~ 3 Y9f[U, Y, 1C 1(a) -+- I] ~ VY d[U, Y, a, wn] ~ 3 Yg(J[U, Y, a, (iJn].

Since 1C 1(a) <wn' we also have 1C 1(a)-+- 1 «iJn'. Therefore by (5) we obtain


Since n l (a) -< wn' occurs as a premise of an implication in !?J [U, Y, a, (ll] we obtain

(6.1) DAf-VYd[U, Y,a,(bn]-dY!?J[U, Y,a,wnl


By the definitions of d and !?J we obtain

DA f- !?JIU, V, a, (bn] - d[U, W, a, (bnl

By Theorem 20.8 using (S2.1) we obtain

(6.2) DAHY!?J[U, Y,a,(bn]_VYd[U, Y,a,wnl

From (6.1) and (6.2) we obtain

(7) DAf-Vx(VYd[U, Y,x,(bn]~-dY!?J[U, Y,x, (bn]).

Thus

j1[UJ: =).xVYd[U, Y, x, (bn]

is a L1 ~-predicator. By Theorem 20.6 using (6.2) we get

DA f- VYd[U, Y, n(a, b), (bn'] _ d[U, V, n(a, b), (bn]

DA f- !?J[U, V, n(a, b), (bn] _ VYd[U, Y, n(a, b), (bnl

Using basic inferences (S4) we obtain

DA f- j1[U](a, b) _ d[U, V, n(a, b), (bn']

DA f- !?J[U, V, n(a, b), (bn] - j1[U](a, b).

From the definitions of d and !?J we obtain

(8.1) DA f- a -«bn' - ~[U, V, a+- I] - j1[U](a, b) - V(a, b)

(8.2) DA f- a -< (bn' - ~[U, V, a +- I] - V(a, b) - j1[ U](a, b).

Further

(8.3) DA f- c -< (bn' _ a -< c _ a -< (bn'

(8.4) DAf-~[U, V,c]-a-<c-~[U, V,a+-I] hold.


From (8.1)-(8.4) we obtain

DA r c -< (h"' -4 9l[U, V, c] -4 \Ix < c\ly(V(x, y) ..... p[ U](x, y)).

Using the Extensionality Theorem 20.9 we obtain

DA r c <fif' -4 9l[U, V, c] -4 ~[U, p[U], c].

By Theorem 20.8 we obtain

DA r c«b"' -4 3Y9l[U, Y, c] -4 ~[U, p[U], c].

Using (5) we obtain

DA r \lxo < (b"'9l[U, p[U], xo]. Then

DA r ~[U, p[U], (b",]

also holds since (b"' is a limit term. Since p[ U] is a LI ~-predicator,

DA r 3 Y9l[ U, Y, (b"'] follows by Theorem 20.6.

Using a basic inference (S2.1) the assertion

DA r \lX3Y~[X, Y, (b",] follows for n'.

theorem 21.5. DA r .Jf[a] holds for every ordinal term rt. < cjJwO.

Proof Suppose rt. < cjJOJO. Then there is a numeral n such that a < $(n, 0). By Lemma 10 we have

DA r \lX3Y9i[X, Y, (b",] -4 ron < (b"' -4 .Jf[wn] -4 .Jf[$(n, 0)].

By Lemma 12 DA r \I X3 Y9l[ X, Y, (b",] holds. Further ron = (b" < (bn' and by Theorem 21.3 DA r .Jf[(b"] also holds. Using cuts DA r .Jf[$(n, 0)] holds. Since a < <p(n, 0), DA r .Jfta] follows by (.Jf3).

7. Deductions in DA '"

As previously, we put

d[U, V,n,m]:==1r 1(n)<m-49l[U, V,1t 1(n)-+I]-4 V(n)

.?I[U, V, n, m]: = 1t 1(n) <m -4 (9l[U, V, 1t 1(n) -+ IJ 1\ V(n)).


Lemma 13. For every ordinal term IX of OT and every numeral n

Proof Let ~[U, V, W, s, t] be the formula given in Lemma 8. From Lemmata 8 and 3 we have

(1) DA*~w2lfX<P<§[U, V, W,x,t]-<§[U, V, W,p,t]

for every numerical term t. We prove

by induction on {3. For {3o < {3, 2· {3o + 1 < 2· {3. Therefore by I.H.

DA*~W2+2'P<§[U, V, W,n,t]

for every numeral n <po By Lemma 4 we have

U sing a cut with (l) the assertion

follows, that is

(2) DA* f£!2+ Z 'P+l P <t- 9l[U, V, t] --+ 9l[U, W, t] - lfy(Vp(Y) - Wp(y».

- -- --

{3 < {3 + 1 and lfy(Vp( y) +-+ Wp( y» - V({3, 1tz(n» - W({3,1t2(n»

are finitely deducible formulas.

Using (2) for t: = {3 + 1 by two cuts we obtain

-

- W({3,1t2(n».

For 1C 1(n) = {3 < OJ·a we have 2· {3 + 3 < OJ ·a. It follows that

for 1C 1(n) <OJ·a. We obtain


for 1tl (n) -( W· a as a logical consequence of (3). (4) also holds for w· a ~ 1tl (n), since the formula concerned is in this case an (Ax2). By Theorem 20.16 and a basic inference (S2.1) we obtain

DA* ~C02+co." 3 Yr!I[U, Y, n, w·a] ~ 'v'Yd[U, Y, n, w·a]

for every numeral n.

Lemma 14. For every ordinal term w·a+n o/OT where n<w

DA* j.!22+ co ',,+n+9 3Y9l[U, Y, w'a+n] holds.

Proo/by induction on w·a+n. For a=O the assertion follows from Lemmata 12 and 13. We may therefore assume a;i:O.

Case 1. n=no+ 1. An elementary predicator j't[V] may be defined such that the following hold:

DA I- 'v'x -( W· a + no 'v'y(j't[ V](x, y) - V(x, y»

DA I- 'v'y(j't[V](w·a+no, y) - 'v'z(h(w·a+no)~ z -(w·a+no

~ 9'[Vz' ¢(e(w·a+no),y)]»·

With the aid of the Extensionality Theorem 20.9 we then obtain

DA I- 9l[U, V, w'a+no] ~ 9l[U, j't[V], w·a+n].

By Theorems 20.6, 20.8 and Lemma 3 we obtain

DA* ~co2 3Y9l[U, Y, w·a+no] ~ 3Y9l[U, Y, w·a+n].

By I.H.

The assertion follows by a cut.

Case 2. n=O. From the definitions of d and fJ4 we have

21. Deductions of Transfinite Induction

By Theorems 20.6, 20.8 and Lemma 3 we obtain

DA* ~W2 3Y9f[U, Y, n1(n)-t- I] ~ VYd[U, Y, n, w·o:]

~ 3Y~[U, Y, n, w·o:].

195

For n1(n)={3 -<w·o: we have n1(n)-t-1 =13+ 1 and P+9<w·0:. Hence by the I.H. we have

and consequently also

for n1 (n) -< w·o:. This also holds for W· 0: ... n1 (n), since the formula

VYd[U, Y,n,w·o:]~ .?l[U, V,n,w·o:]

is in this case an (Ax2). Hence (1) holds for every numeral n. Using Lemma 13 we obtain

DA* ~w2+w.a VYd[U, Y, n, w·o:] <--> 3Y~[U, Y, n, w·o:].

Using a basic inference (S2.0*) we obtain

By the definitions of d and ~ we obtain from the formulas

VYd[U, Y, n(m, n), w·o:] ~ d[U, V, n(m, n), w·o:]

~ [U, V, n(m, n), w·o:] ~ 3Y~[U, Y, n(m, n), w·o:]

which are deducible in DA, that

DAf-m-<w·o:~9f[U, V,m-t-l]~VYd[U, Y,n(m,n),w·o:]~ V(m,n)

DA f- m -<w·o:~ 9f[U, V, m-t-l] ~ V(m, n)~ 3Y~[U, Y, n(m, n), w·o:].

Using Lemmata 3 and 13 we obtain

DA* ~w2 m -<w·o:~ 9f[U, V, m-t-l] ~ VYd[U, Y, n(m, n), w·o:] ~ V(m, n)

DA* ~w2+W"a m -<w·o: ~ 9f[U, V, m-t- 1] ~ V(m, n)

~ VYd[U, Y, n(m, n), w·o:].


Using a basic inference (S4) for the predicator

fi[U]: =hVY..#[U, Y, x, w·cx]

we obtain as a logical consequence

Using a basic inference (S2.0*) we obtain

For k -<w·cx the formulas m -<k -4 m -<W·L/. and \

~[U, V, k] -4 m -<k -4 :f4'[U, V, m+ I]

are deducible in DA and therefore in DA * with orders < w 2 . Therefore using two cuts with (3) we obtain

DA* r 2 +ro-a+2 ~[U, V, k] -4 m -<k-4 Vy(V(m, y) +-+ fi[U](m, y))

and by a basic inference (S2.0*)

(4) DA * r 2 + w "+ 3 .~[ U. V. k] -. IIx -< klly( Vex. y) <-> A U](x. y))

for k -< W· CI.. With the aid of the Extensionality Theorem 20.9 the formula

~[U, V. k] -4 IIx < klly( Vex. y) <-> A U](x . .1')) - ~jf[ u. AU]' k]

is deducible in DA and therefore in DA * with order < 0/. Therefore by a cut using (4) we obtain

DA* r 2 +ro-d4 ~[U, v. k] -4 .~[U. AU]. k] for k <w·cx.


DA* r 2 + w 'd5 3Y~[U, Y, k] -4 ~[U, fi[U], k].

By I.H. we l;lave

DA* ~w2+w.a 3Y~[U, Y, k] for k -<w·cx. It follows that

DA* r 2 + W ' d6 ~[U, fi[U], k] and by Lemma 4

DA * r 2 + W ' d7 Vx <w ·cx ~[U, fi[U], xl

22. The Semi-Formal System RA* for Ramified Analysis 197

The formula

\Ix -< w·a ~[U, .f[U], x] --> ~[U, .f[U], w ·a]

is deducible in DA and therefore in DA* with order <w2 . Hence we obtain

(5) DA * f!£2+ on+8 ~[U,.f[ U], w . a].

By Theorem 20.17 the assertion

DA* f!£2 +W"a+ 9 3Y~[U, Y, w·a] follows from (2) and (5).

Theorem 21.6. DA* fb+13 .. IHn+l]for n~ 1.

Proofby induction on n. By Lemmata 10 and 3 we have

w 2 + (n = (n for n ~ 1. Therefore by Lemma 14 we obtain

DA* fb+l0 3Y~[U, Y, (n+ 1]. Using (S2.1) we obtain

DA* fb+ll \lX3Y~[U, Y, (n+ 1]. Bya cut with (1) we obtain

(2) DA* fb+12 ,j'T''Zn] --> f[(n+l].

Ifn= 1, then (n=eo<<pwO. Then by Theorem 21.5 and Lemma 3 DA* ~ro2 f[~n]. By the I.H. in every case we have DA * ~Sn f[~n]. Using a cut with (2) we obtain DA* fb+13 f[(n+l].

§22. The Semi-Formal System RA * for Ramified Analysis


We use the following as primitive symbols: 1. Denumerably infinitely many bound number variables. 2. Denumerably infinitely many free (I-place) predicate variables of level a

for each ordinal term a of the system ~T.


3, Denumerably infinitely many bound (I-place) predicate variables oflevel /3 for each ordinal term /3#0 of the system ~T.

4. The symbols 0, " ..l,~, V and A. 5. Symbols for n-place calculable arithmetic functions and n-place decidable

arithmetic predicates (n ~ 1). (The conditions specified on p. 168 for the use of these symbols have to be satisfied.)

6. Round brackets and comma. In this section small Greek letters (possibly with indices) always denote

ordinal terms of the system ~T. We use nominal forms in the usual way and denote them by capital script

letters. Terms, numerals and prime formulas are defined as in §20.1. In the formal language of the system RA * every term is numerical and every prime formula constant. Therefore by §20.I we have: Every term has a calculable value and every prime formula is decidably true or false.

Inductive definition of formulas and predicators: 1. Every prime formula is a formula of level O. 2. Every free predicate variable of level ct is a predicator of level ct.

3. If pa is a predicator oflevel ct and t is a term, then paCt) is a formula oflevel ct.

4. If A and B are formulas of levels ct and /3 then (A ~ B) is a formula of level y: = max (ct, /3).

5. If $'[0] is a formula of level ct and x is a bound number variable which does not occur in $', then Vx$'[x] is a formula oflevel ct and A.x~[x] is a predicator of level ct.

6. If UfJ is a free predicate variable oflevel /3#0, $'[UfJ] is a formula oflevel ct and X fJ a bound predicate variable of level /3 which does not occur in $', then VXfJ$'[XfJ] is a formula of level y=max(ct, /3).

Corollaries. 1. The level of a formula is 0 if it contains no predicate variables, otherwise it is the maximum of the levels of the predicate variables which occur in it.

2. If s, t are terms then: If $'[s] is aformula, $'[t] is also aformula. 3. If pa is a predicator of level ct, $'[pa] aformula and QfJ a predicator of level

/3 which contains no bound variable occurring in $', then $'[QfJ] is also a formula.

Syntactic symbols: x, y, z for bound number variables, Ua, V", wa for free predicate variables of level ct, XfJ, yfJ, ZfJ (where /3#0) for bound predicate variables of level /3, P", Q" for predicators of level ct,

s, t for terms, A, B, C, D, E, F, G for formulas, i, j, k, m, n for natural numbers and the corresponding numerals. We shall also use these syntactic symbols with subscripts.

In the same way as before we write A ~ B for (A ~ B) and -,A for (A ~ ..l). We also denote an ordinal term (1)o·n by n for short. As before, the length of a formula is the number of~, V and A symbols, which occur in it.


A predicator P~ is said to be:F -permissible if :F[P~] is a formula. In particular this is the case when :F[UO] is a formula and P~ contains no bound variable occurring in :F.

We now assign an ordinal term of the system OT as a rank to each formula.

Inductive definition of the rank of a formula: 1. Every prime formula has rank o. 2. Every formula U~(t) has rank p: = w· <X.

3. If A and B are formulas of ranks PI and P2 then the formula (A ~ B) has rank p: = max (PI' P2)+ 1.

4. If :F[O] is a formula of rank Po in which x does not occur then the formulas 'v'x:F[x] and h:F[x](t) have rank p: = Po + 1.

5. If :F[UO] is a formula of rank Po and 'v'XP:F[XP] is a formula of level y then [UO] has rank p: = max (w·y, Po+ 1).

Corollaries. 1. Two formulas :F[s] and :F[t] have the same rank. 2. Two formulas :F[ U~] and :F[ V1 have the same rank.

Theorem 22.1. Every formula of level y has rank P = w· y + n (where n is an abbreviation for WO . n and therefore P < W· Y + w).

Proof By induction on the length of the formula.

Theorem 22.2. If P~ is an :F-permissible predicator of level <x<y and :F[UO] is a formula of level y then :F[ P~] has the same rank as :F[ UO].

Proof by induction on the length of the formula :F[UO]. We may assume that :F[P~] and :F[UO] are different formulas since otherwise the assertion is trivial. It then follows since y # 0 that :F[ UO] has length > O.

1. Suppose :F[UO] is a formula :FI[UO]~:F2[UO] or 'v'~[UO,x] or A~[UO, x](t). Then the assertion follows from Theorem 22.1 and the I.H.

2. Suppose:F[UO] is a formula 'v'Xp~[UO, XP]. Then ~[UO, V 9 ] is a formula oflevel yo~y. Ifyo=y then by I.H. ~[UO, VO] and ~[P~, VO] have the same rank Po. In this case both formulas :F[UO] and :F[P~] have rank Po+ 1. If Yo<y then ~[P"', VO] is also a formula of level <yo In this case both formulas :F[UO] and :F[P"'] have rank w·y.

Theorem 22.3. a) A and B have smaller ranks than A ~ B. b) :F[t] has smaller rank than 'v'x:F[x] and AX:F[X](t). c) Every formula :F[P"'] where rx<P has smaller rank than 'v'XP:F[XPl

Proof a) and b) follow immediately from the definition of the rank of a formula. c) is obtained as follows: Ify is the level of the formula 'v'XP:F[XP] then :F[UO] is a formula of a level Yo'" y. If yo=y, then by Theorem 22.2 :F[UO] and ~[P"'] have the same rank Po while the formula 'v'XP:F[XP] has rank Po+ 1. If Yo<y then :F[P"'] is also a formula ofa level <yo By Theorem 22.1 it has a rank <w·y while the formula 'v'XP:F[XP] has rank w·y in this case.


2. The Deduction Procedures

In the same way as before we define positive and negative parts of formulas, FP- G and the equivalence offormulas. We also use the same syntactic symbols as before for P-forms, N-forms and NP-forms.

Axioms of the system RA* (corresponding to those for DA*): (Axl) &l[A], if A is a true prime formula. (Ax2) JV[A], if A is a false prime formula (Ax3) ,q[A, B], if A and B are equivalent formulas of length O.

We call the minimal parts indicated in the axioms the principal parts of these axioms.

Basic inferences of the system RA * : (SI) JV[iA], JV[B] f-JV[(A ~ B)], if B is not the formula-1.. (S2.0*) &l[$'[n]] for every numeral n f- &l[\fx$'[x]]. (S2.1 *) &l[$'[pa]] for every $'-permissible predicator pa of a level

!X<f3 f- &l[\fXfJ$'[XfJ]]. (S3.0) $'[t] ~ JV[\fx$'[x]] f- JV[\fx$'[x]]. (S3.1) $'[pa] ~ JV[\fXfJ$'[XfJ]] f- JV[\fxfJ] $'[xfJ]] , if !X<f3. (S4) $[$'[t]] f- $[h$'[x](t)]. (Cut) &l[A], A ~ B f- C, if &l[B] P- C holds.

All premises and all conclusions of these basic inferences must be formulas. The basic inferences (S2.0*) and (S2.1 *) have infinitely many premises. Every other basic inference has either one or two premises. We call the basic inferences (S 1 )-(S4) principal inferences. We call the minimal part indicated in the conclusion of a principal inference the principal part of that inference. The formula denoted by A in the premises of a cut is called the cut formula. The rank of a cut is the rank of its cut formula.

Inductive definition of RA * f} F 1. If F is an axiom of the system RA * then RA * f} F holds for all ordinal terms IX and p of the system OT.

2. If RA * W Fi and !Xi <!X for every premise Fi of a principal inference or a cut of rank < p then RA * f} F holds for the conclusion F of that inference.

Corollary. If RA * f} F, !X ~ f3 and p ~ (7, then RA * ~ F also holds.

A formula F is said to be deducible in RA * with order !X and rank p if RA * f} F holds. Thus !X is an upper bound for the orders of the basic inferences which occur in the deduction of F while p says that every cut which occurs in the deduction of F has a rank < p. If RA * f1r F holds then the formula F has a cut-free deduction with order !x.

Predicativity of the system RA *. On the basis of the intended interpretation of the formulas and predicators every formula and every predicator which contains a


universal quantifier V X fJ depends on the totality of those predicators which are to form the domain of this universal quantifier. The basic inferences (S2.1 *) and (S3.1) of the system RA * are so formulated that the domain of a universal quantifier V X fJ consists only of predicators of level < {3 while every formula and every predicator which contains the universal quantifier V XfJ belongs, according to the definition, to a level ~ {3. Therefore by means of the levels we have given a predicative construction of our formal language. We shall also show on pp. 217-220 that the transfinite induction over ordinal terms of the system OT which we use here is predicatively provable. Thus not only the formal language but also the deduction procedures of the system RA * are seen to be predicative.

3. Weak Inferences

Below we write ~ F for RA * I%- F. F f- G is said to be a weak inference if the following always holds: If I%- Fthen ~ G.

Theorem 22.4 (Replacement rule). If F and G are equivalent formulas F f- G is a weak inference.

Proof. In this case I%- G follows from ~ Fby induction on rJ..

Theorem 22.5 (Inversion rules). The following are weak inferences:

a) %[(A -> B)] f- %[iA]

b) %[(A -> B)] f- %[B]

c) g>[Vx~[x]] f- g>[~[t]]

d) g>[VXfJ ~[XfJ]] f- g>[~[pa]], if pa is ~-permissible and rJ.<{3.

e) C[AX~[X](t)] f- C[~[t]].

The inductive proofs proceed as for Theorem 4.3.

Theorem 22.6 (Structural rule). If Ff1- G holds then Ff1- G is a weak inference.

Proof. Using Theorem 22.5, as for Theorem 4.4.

4. Elimination of Cuts

Lemma 1. If A is a true prime formula and I-tr A -> B holds, then I-tr g>[ B] for every P-form g>.

Proofby induction on ex. 1. Suppose A -> B is an axiom. If A is not a principal part of this axiom, then

g>[B] is an axiom of the same sort. Otherwise B is a formula g>1[A 1], where Al


is also a true prime formula. Then &,[B] is the (AxI) &'[&'1[A 1]]. In each case 1-0- &,[B] follows.

2. Suppose A -) B is the conclusion of a principal inference. Then the assertion follows from the I.H. using the structural rule.

Lemma 2. If A is a false prime formula and 1-0- &,[A] holds, then 1-0- &,[B] for every formulaB.

Proofby induction on IX.

1. Suppose &'[A] is an axiom. If A is not a principal part of this axiom then &,[B] is an axiom of the same sort. Otherwise &,[A] is an (Ax3) ,q[A1' A] where A1 is also a false prime formula and &,[B] is the (Ax2) ,q[A1' B]. In each case 1-0- &,[B] follows.

2. Suppose &'[ A] is the conclusion of a principal inference. Then the assertion follows immediately from the I.H.

Lemma 3. If A is a formula Urt(t) or Vxff[x] or VXPff[XP] of rank p and It &,[A] andf% A -) B hold, then ItH &,[B].

Proofby induction on fJ. 1. Suppose A -) B is an axiom. If &,[B] is also an axiom then the assertion is

trivial. Otherwise A is a formula Urt(t) and B is a formula &'1 [Urt(s)] where sand t are terms of equal value. Then It H &'[&'1 [U"(s)]] follows from It &'[ Urt(t)] by the replacement rule and the structural rule.

2. Suppose A -) B is the conclusion of a principal inference with principal part A. Then A is a formula Vxff[x] or VXPff[XP] and we have fJo<fJ such that

where, in the second case, IX < p. By the structural rule and the I.H. we obtain

In this case by the inversion rules, from It &,[A] we obtain

It &,[ff[t]] or It &'[ff[P1]'

Then ff[t] or ff[Prt], respectively, has a rank <p by Theorem 22.3. Therefore ItH &,[B] follows by a cut with cut-formula ff[t] or ff[Pa:J.

3. Suppose A -) B is the conclusion of a principal inference whose principal part is in B. Then the assertion follows from the I.H. by the structural rule.

4. Suppose A -) B is the conclusion of a cut &'1 [C], C -) D I- A -) B such that &'1 [D] p.. A -) B, whose cut-formula C has a rank < p. From the formulas &'l[C] and D we obtain formulas &'o[C] and Do such that


by either removing or introducing negative parts A in such a way that f!J' o[Do] f!- B holds. Now there exist 15;<15 (i= 1,2) such that

f%t &UC] and 1}2 C- D.

By the structural rule we obtain

By I.H. and the structural rule we" obtain

If~' g}'[g}'o[ C]] and ~H2 C - g}'[Do].

Bya cut with cut-formula C of rank <p we obtain ~H g}'[B], since g}'[f!J'o[g}'[Do]]] f!- g}'[B] holds.

Lemma 4. 1;+ 1 F implies ft-0a F.

Proofby induction on IX. If F is an axiom then the assertion is trivial. Suppose now that F is the conclusion of a basic inference BI. Then there exist IX; < IX such that It' + 1 F; for every premise F; of BI. Hence by I.H. ft- 0a, F;.

1. Suppose BI is a principal inference or a cut of rank <po By BI we obtain I:*0a since 4JOIX; < 4JOIX for all IX; < IX. "

2. Suppose BI is a cut g}'[A], A - B f- F such that ,?l'[B] f!- F where the cutformula A hasrankp. Then Wal g}'[A] and ft-0a2 A - Bhold where lXi <1X(i= 1,2).11 suffices to prove ft-0a g}'[B] since then ft-0a F follows by the structural rule.

2.1. Suppose A is a prime formula. Then p = ° and A is either true or false. In either case ft-0a g}'[B] follows by Lemmata 1 and 2.

2.2. Suppose A is a formula Uao(t) or 'v'xff[x] or 'v'XPff[XP]. In these cases wa g}'[B] follows by Lemma 3 since 4J01X 1 + 4J01X2 < 4JOIX.

2.3. Suppose A is a formula Axff[x](t). Then by the inversion rule we have Wal g}'[ff[t]] and ft-0a2 ff[t] _ B. By Theorem 22.3 ff[t] has a rank <po Therefore by a cut with cut-formula ff[t] we have wa g}'[B].

2.4. Suppose A is a formula Al - A2. Then by Theorem 22.3 Al and A2 have ranks <po From ft-0a2 (Al - A2) - Bbythe inversion rule we have ft-0a2 ,A1 - B and ft-0a2 A2 _ B. Using ft-0al g}'[(Al - A2)] we obtain ft-0ao + 1 Al - g}'[B] by a cut with cut-formula A2 and by the structural rule where 1X0: =max (IX 1 , 1(2). Using ft-0a2 ,A1 - Bweobtain ft-0a g}'[B] by acutwithcut-formulaA 1 , where4JOlX2 < 4JOIX, 4JOlXo + 1 < 4JOIX and ,g}'[ B] - B f!- g}'[ B].

Theorem 22.7 (First Cut Elimination Theorem). I; F implies fta F.

Proof by induction on p with a subsidiary induction on IX. If F is an axiom the assertion is trivial. Suppose now that F is the conclusion of a basic inference BI. Then there exist lXi < IX such that ~ F; for each premise Fi of BI. By the subsidiary I.H. we have Ita, F i •


1. Suppose BI is a principal inference. Then by BI we have ~P" F since ¢PlY.i < cPPIY. for alllY.i<lY.·

2. Suppose BI is a cut of rank Po < p. Using this cut we obtain Ita~ 1 F. Since polO, cPO(cPPIY.) = cPPIY.· Therefore by Lemma 4 Wo'" F. Since Po<P we also have cPPo(cPPIY.) = 4!PIY.· Therefore by the main I.H. we obtain W'" F.

Theorem 22.7 can be sharpened following W. Tait [3J thus:

Theorem 22.8 (Second Cut Elimination Theorem). It+wv F implies I-$v", F.

Proofby induction on v with a subsidiary induction on IY.. The assertion holds for v = 0 by Lemma 4. If F is an axiom the assertion is trivial. Now suppose v # 0 and F is the conclusion of a basic inference BI. Then there exist lY. i < IY. such that W +wv Fi for each premise Fi of BI. By the subsidiary I.H. we obtain I-$v"" Fi .

I. Suppose BI is a principal inference or a cut of rank < /3. Then using BI we have I-$v", F.

2. Suppose BI is a cut of rank P where {3 ~ P < {3 + WV. Then there exist Vo < v and nE N such that p<{3+wvo ·n. Using this cut we obtain l-$':wvo.nF. Since Vo < v, 4lvo(cPvlY.) = cPwY.. Therefore by n-fold application of the main I.H. we obtain W"'F.

5. Fprther Properties of Deductions

Theorem 22.9. /fC l and Cl are equivalent formulas of rank p, then ~.p 22[C l , Cll

Proof by induction on P as in the proof of Theorem 4.1. For example, if Ck is a formula \fXPffk[XPJ and F; is an ffl-permissible predicator, then Pi" is also g>lpermissible and by Theorem 22.3 g>k[Pt] is a formula of rank Pi < p. Then by I.H.

Hence by basic inferences (S3.1) we obtain

and using a basic inference (S2.1 *)

since 2· Pi + 1 < 2· P holds for all Pi < p.

Theorem 22.10. /fO<{3<y and\fXYg>[XYJ is aformula of rank p, then


Proof By Theorem 22.9

holds. By an inversion rule we obtain

for every ff -permissible predicator pa of level rx. < p. Hence the assertion follows by a basic inference (S2.1 *).

Definitions. As before we set

(A +-+ B): = -,«A - B) - -,(B- A» 3XP ff[ XP] : = -, If xP -, ff[ XP].

Theorem 22.11 (Extensionality Theorem) . .if ff[A.xd 1 [x]] and ff[A.xd z[x]] are formulas of maximal rank p then

holds for (i,j)=(I, 2) and (i,j) =(2, I).

Proofby induction on p. For brevity we set

A: = Ifx(d 1 [x] +-+ .91 2[X]),

Pk :=2xdk [x] (k=I,2).

The assertion then reads

l. Suppose ff[P;] has length O. Then Pi does not occur in ff[P;] so ff[P 1]

and ff[Pz] are the same. In this case the assertion holds by Theorem 22.9. 2. Suppose ff[P;] is a formula Pi(t). Then by Theorem 22.3 the formulas

d1(t) and .912(/) have a maximal rank Po<p. We set

%:={(*l- -,(d2[t]-d1[t]» if(i,j)=(1,2),

«.91 1[t] - .91 2[t]) - -, * 1)' if (i,j) =(2, I).

Then % is an N-form such that -, %[(di[l] - dJt])] is the formula (.91 1[t] +-+ .91 2[t]). By Theorem 22.9 we have

f-ff·PO -,%[-,.91;[1]] - (A - ,q[d;[l], dlt]]),

f-ff·po -,.;V[dJt]] - (A - ,q[d;[l], dltJ]).


Hence by a basic inference (S 1)

and by a basic inference (S3.0)

since 2· Po + 1 < 2· p. The assertion follows by two basic inferences (S4). 3. Suppose%[P;J is a formula %1 [p;J ---> %2[P;J odx'§[Pi, x](t) or \fx'§[Pi ,x]

or \fXP,§[pi'XP]. Then the assertion follows from the I.H. by appropriate basic inferences.

Theorem 22.12. Ij\fY"id[n, Y"i] and3 Y"i,i3l[n, Y"i] areformulasoflevelai{i = 1,2) where O<a l <a2 and

~\fX(\fY"i d[x, y"i]~-dY"i,i3l[X, Y"i]) (i=1,2)

holds where w· a 2 + w!( It. and w . a 2 + w!( p then

~+5 \fx(\fY"'d[x, Y"'] _ \fY"2d[x, Y"2]).

Proof For brevity we set

dJn]:=\fY"id[n, Y"i] and ,i3li[n]:=3Y"i,i3l[n, Y"i] (i=I,2).

Let the maximal rank of the formulas dJn], {!BJn](i= 1, 2) be Po. By Theorem 22.1 po<w·az+w. Then 2·po<w·a2+w also and therefore 2·po<1t. and Po<p. From the hypotheses I-} \fx(dJx] - {!BJx])(i= 1, 2} by the inversion rule we obtain

that is,

~ --, ((dJn] ---> {!B In]) ---> --, (,i3l In] ---> dJn])).

Hence by the inversion rules we obtain

Using the axioms -1---> {!Bl[n] and -1---> d 2[n] we obtain by cuts with cut-formula -1 and rank O<p

~+l--,(dl[n]---> {!Bl[n])---> {!Bl[n] and ~+l--,({!B2[n]--->d2[n])

---> d zen].


By the structural rule we obtain

The formula \fyu2,~[n, Y U2] has a rank <Po. Since 2·po<rt.. it follows by Theorem 22.10 that

that is

(2) I%- ~ len] ---+ &l 2[n].

The formulas &l1[n] and ~ 2[n] have ranks <po Therefore by two cuts from (1) and (2) we obtain

By Theorem 22.10 we also have

From (3) and (4) by the structural rule we obtain

Using a basic inference (Sl) we obtain

for every numeral n. The assertion f%+ 5 \fx(d I [x] ~ d 2[X]) follows by a basic inference (S2.0*).

6. Interpretations of EA * and DA * in RA *

If F is a formula of the system DA * and 0' # 0 then we denote by FU a formula of RA * which results from F by replacing every bound predicate variable X by XU and every free predicate variable by a predicator of a level < 0'. Then F U is a formula of a level ~ 0'. It has a rank < w· 0'+ W.

For the subsystem EA * of DA * which lacks the basic inference (S3.L1) we have:

Theorem 22.13 (First Interpretation Theorem). EA * ~ F implies RA * H;;!~ Fl.


Proof by induction on IX. Suppose EA * ~ F holds. 1. Suppose F is an axiom of the system EA *. If F is an (Axl) or (Ax2) then

the assertion holds since then Fl is an axiom of the system RA *. If F is an (Ax3) then Fl is a formula .El[Cl, C2] where Cl and C2 are equivalent formulas oflevel O. They have a rank <w. Thus the assertion holds by Theorem 22.9,

2. Suppose F is the conclusion of a basic inference BI of the system EA"'. 2.1. Suppose BI is a basic inference (S2.1). Then F is a formula ,o/I[\f X g;[ X]]

and there exists lXo < IX such that

EA * ~o ,o/I[g;[ U]],

where U does not occur in F, Fl is a formula ,0/11 [\f Xl g;l [Xl]]. By I.H.

holds for every g;l-permissible predicator Po of level O. The assertion follows using a basic inference (S2.1 *).

2.2. Suppose BI is not a basic inference (S2.1). If BI is a cut with cut-formula A then BI is replaced by a cut with a cut-formula Al of level ~ 1 whose rank is < w + w. In every case the assertion follows immediately from the I.H.

Theorem 22.14 (Second Interpretation Theorem). DA* ~ F(or DA * f1- F) implies

RA * H;;: ~!:+6' a F' for any a= w a , f3 (or a = 2~· f3) such that f3 #0.

Proof by induction on IX. Suppose DA * ~ F (or DA * If F) holds and a = wa • f3 (or a=2a • f3) where f3=10.

I, Suppose F is an axiom of the system DA *. If F is an (Ax!) or (Ax2) then the assertion holds since then F" is an axiom of the system RA *. If F is an (Ax3) then pI is a formula .El[Cl , C2 ] where Cl , Cz are equivalent formulas of a level < a, They have a rank p < w· a. Then 2· p < w· a also. Therefore the assertion holds by Theorem 22.9.

2. Suppose F is the conclusion of a basic inference BI of the system DA * which is not an inference (S3.M. Then there exist lXi < IX such that DA * ~ Fi (or DA * IT F) for each premise Fi of BI. Putting f3i: =w -ai+a. f3 (or f3i: = 2- ai +a. f3) we have f3i =10 and W lli . f3i = wa. f3 = a (or 2~i. f3i =2a. f3 = a). Therefore by the I.H.

holds for appropriate formulas F;". The assertion follows as in the proof of Theorem 22. B.

3. Suppose F is the conclusion of a basic inference (S3 . .1) of the system DA *. Then F" is a formula fi[\fX"g;[X"]] and by I.H. there exist lXi<1X (or lXi+W~IX) (i = 1, 2) such that

(1) RA* H;;:~!:+6.a, \fx(\fY"d[x, Y"] ~-dY"gj[x, Y"]),

(2) RA*H;;:~!:+6.a2g;[Ax\fY"d[x, Y1]~fi[\fX"g;[X"]],

23. The Limits of the Deducibility of Transfinite Induction 209

where den, VO] and fJ6 [u, VO] are formulas of a maximal level 0"0<0". Since a l <cx and O"=w",p (or a l +w~a and 0"=2~·P), by Theorem 14.12 (or 14.13) there is an ordinal term 0"1 = w~, . Pl (or 0"1 "'" 2'" . Pl) such that 0"0 < 0" 1 < 0". By Theorem 22.11

(3) RA'" 1iQ.,,+w{V'X(V'Y"'d[X, Y"'] +-+ V'Y" d[x, P]) m _ (ff[AXV'P'd[x, Y"']] _ ff[hV'Y" d[x, P]] holds.

By a cut from (2) and (3) we obtain

RA*lw',,+w+6'a2 +l ' , {ff[AXV'Y t1 'd[X Y"']] _ (V'x(V'Y"'d[x Y"']

r,;;·,,+w +-+V'P d[x, Y"]) _ %[V'X"ff[X"]).

Here hV'Y"'d[x, Y"'] is an ff-permissible predicator of level 0"1 <0". Therefore using a basic inference (S3.1) we obtain

(4) RA* IW,,,+w+6.a2+2{V'X(V'Y<1'd. [x, P'] +-+ V'Y" d[x, Y"]) . r,;;',,+w _ %[V'X"ff[X"]].

By I.H. we also have

(5) RA"'H;;:::~~+6''''VX(V'P'd[x, Y<1']+-+3Y"'fJ6[x, YO',])

From (1) and (5) by Theorem 22.12 we obtain

and we have 6·cx l +S<6·a and 6·cx2 +2<6·a. Therefore by a cut from (4) and (6) we obtain the assertion

§23. The Limits of the Deducibility of Transfinite Induction

1. Orders of Deductions of Induction in RA '"

As in §21 we define in RA"':

fl1t[P"J: = V'X2(V'X l -<X2P"(X 1) - P"(X2))

JP[t]: = V' XP(fl1t[XP] - V'x -< tXP(x)).

Lemma 1. Assume: I. M + and M _ are diSjoint finite sets of numerals, M + is non-empty and

}' : = min {t(n) I n E M + }.


II. F is aformula with the following properties: The minimal negative parts of Fare .?h[U"'J, all U"'(n) such that n E M _ and otherwise only true prime formulas (if any). The minimal positive parts ofF are only the U"'(n) such that n E M +.

III. RA* ~F. Then: y<w·c5.

Proofby induction on c5. By hypotheses I and II F can only have been obtained by a basic inference (S3.0). Therefore it follows from hypothesis III that there exist c50 < c5 and a numerical term t such that

Suppose the numerical term t has value m. Then by Theorems 22.4 and 22.5a) and b) from (1) we obtain

(2) RA* I-&' -,\fx1 -<mU"'(x1)---+ F

(3) RA* ~ U"'(m)---+ F.

1. Suppose m ¢ M +. Then the formula U"'(m) ---+ F satisfies hypotheses I and II of our lemma with the same set M + as for F. Therefore from (3) by I.H. y < w· c5 0

and therefore y < w· c5. 2. Suppose mEM+. Then y~r(m). Since M_ is a finite set and c50 <c5 there

exists a numeral k¢M_ such that w·c5o~r(k)<w·c5. By Theorem 22.5c) from (2) we obtain

Assume that r(k)<y. Then k-<m is a true prime formula and the formula -,(k -<m ---+ U"'(k)) ---+ F satisfies hypotheses I and II of our lemma with {k}uM + instead of M + and r(k) = min {r(n) In E {k}uM +}. By I.H. from (4) it follows that r(k) < w· c50 contradicting the choice of k. Thus the assumption r(k) < y is refuted. Hence y ~ r(k) whence y < w· c5.

Proof By Theorem 22.5c) from the hypothesis we obtain

By Lemma I we have ~<w·c5 for all ~<y, therefore y~w·c5.

Theorem 23.2. RA* ~ JP[n (where P=I=O) implies y~w·c5.

Proof By Theorem 22.5d) from the hypothesis we obtain

RA * ~ &l1[U"'J ---+ \fx -<yU"'(x)

for r:x<p. By Theorem 23.1 we obtain y~w·c5.


2. The Limiting Numbers of the Systems EN, EA and DA

By the limiting number of the system EN we mean the least ordinal term y such that the formula

.?h[U]~ Vx<.YU(x)

is not deducible in EN.

Theorem 23.3. The formal system EN has limiting number Bo.

Proof Suppose

(I) EN I- &'~[U] ~ Vx <.YU(x).

By Lemma 3 of §21 we obtain

(2) EN* pro2 &'~[U] ~ Vx <.yU(x).

EN* can be regarded as a subsystem of RA * in which every free predicate variable belongs to level 0 and every formula has a rank <w. It follows from (1) that the formula

has a deduction in this subsystem with cut-rank restricted to < w. Therefore there exists a numeral n such that

Now cpOW2<Bo and cpOa<Bo for a<Bo. Therefore it follows from (3) by n-fold application of Theorem 22.8 that there exists b < Bo such that

By Theorem 23.1 it follows that y:::;W·b<Bo. Consequently EN has a limiting number :::;Bo. It follows from Theorem 21.1 that Bo is the limiting number of EN.

By the limiting number of the system EA (or DA) we mean the least ordinal term y such that the formula J[y] is not deducible in EA (or in DA).

Theorem 23.4. Theformal system EA has limiting number cplBo.

Proof Suppose

(I) EA I- J[y] holds.


By Lemma 3 of §21 we obtain

(2) EA * ~w2 Jl"[YJ.

By Theorem 22.13 EA* can be interpreted in RA* where every formula has a rank < w + w. It follows from (l) that the formula Jl"l [yJ has a deduction in RA * with cut ranks restricted to < w + w. Therefore there is a numeral n such that, by Theorem 22.13,

It follows from (3) by n-fold application of Theorem 22.8 that there exists <5 < So

such that

Hence by Theorem 22.8

By Theorem 23.2 we have y ~ </J 1<5 < </J Iso. Consequently EA has a limiting number ~</Jlso. It follows from Theorem 2l.3 that </Jlso is the limiting number ofEA.

Theorem 23.5. The formal system DA has limiting number </JwO.

Proof Suppose

DA f- Jl"[yJ

holds. By Theorem 20.13 there exists an ordinal term w·m+n<w2 such that

DA* f.1. rn +n Jl"[YJ.

Putting a:=2w · rn +n =wm ·2" from Theorem 22.14 we obtain

Here w·a+w~w·a+w+6·(w·m+n)<Wm+2. Therefore we also have


RA* ft-(m+2)(W m + 2 ) Jl""[Y].


By Theorem 23.2 we obtain y~w·rP(m+2)(wm+2)<rPwO. Consequently DA has a limiting number ~rPwO. By Theorem 21.5 it follows that rPwO is the limiting number of DA.

3. The Autonomous Ordinal Terms of the Systems EN*, EA* and DA*

Inductive definition of the autonomous ordinal terms of the semi-formal system EN*: 1. If y is less than the limiting number of EN then y is autonomous in EN*. 2. If () is autonomous in EN* and

EN* f! ~,z[U] ~ Vx -< yU(x)

holds where ()<y, then y is also autonomous in EN*.

Lemma 2. If y is autonomous in EN*, then y + 1 is also autonomous in EN*.

Proofby induction on y. If Y is less than the limiting number of EN then y + 1 is also less than the limiting number of EN and therefore autonomous in EN*. Otherwise there is an autonomous ordinal term () < y such that

(1) EN* f! ~,z[U] ~ Vx -<YU(x) .

. We may assume that ()~W2. The formula

~,z[U] ~ Vx-<yU(x)~ Vx-<y+ lU(x)

is deducible in EN. By Lemma 3 of §21 it follows that

(2) EN* ~ro2 ~,z[U] ~ Vx -<yU(x) ~ Vx -<y+ lU(x).

From (1) and (2) by a cut we obtain

EN* f!+1 ~,z[U]~ Vx-<y+1U(x).

Here () + 1 < y + 1 and by I.H. () + 1 is autonomous in EN*. Consequently y + 1 is also autonomous in EN*.

Lemma 3. The autonomous ordinal terms of EN* form a segment. That is: If y is autonomous in EN* and Yo <y then Yo is also autonomous in EN*.

Proofby induction on y. If Y is smaller than the limiting number of EN, then Yo is autonomous in EN*. Otherwise there is an autonomous ordinal term () < y such that

(1) EN* f! 2h[U] ~ Vx -<yU(x).


We may assume that ()~W2. The formula

Vx -<yU(x) -4 Vx -<yoU(x)

is deducible in EN for Yo<Y. It follows by Lemma 3 of§21 that

(2) EN* ~ro2 Vx -<yU(x) -4 Vx -<yoU(x).

By a cut from (1) and (2) we obtain

(3) EN* ~+1 ~h[U] -4 Vx -<Yo U(x).

By Lemma 2 15+ 1 is autonomous in EN*. Consequently Yo is autonomous in EN* by I.H. ifYo~bo+ 1 and otherwise by (3).

Definition. Following S. Feferman [1] we denote by Aut(EN*) the least ordinal term which is not autonomous in EN*.

Theorem 23.6. Aut(EN*) = ¢20.

1. Proof of ¢20~Aut(EN*). Set OCo:=80 and ocn+1:=8~n=¢locn. Then ocn<ocn+1 and w2 +w·ocn =ocn • Therefore from Theorem 21.2 we have

(I) EN* f!£2 &>.z[U] -4 Vx -< aoU(x)

(2) EN* ~n&>.z[U] -4 Vx -<ocn+1 U(x).

By Theorem 21.1 w2 is autonomous in EN*. It follows by induction on n using (1) and (2) that every ordinal term ocn is autonomous in EN*. For each ordinal term y < ¢20 there is a natural number n such that y < ocn. Therefore by Lemma 3 every ordinal term <¢20 is autonomous in EN* and therefore ¢20~Aut(EN*).

2. Proof of Aut(EN*) ~ ¢20. By Theorem 23.3 ¢20 is greater than the limiting number of EN. Suppose

(1) EN* ~ &>.z[U] -4 Vx -<yU(x)

holds, where 15 < ¢20. We have to prove that y is also less than ¢20. We regard EN* as a subsystem of RA * in which every free predicate variable belongs to level 0 and every formula has a rank <w. Then it follows from (1) that


RA* ~1~ &>.z[UO] -4 Vx -<YUO(x).

By Theorem 23.1 we have y~w·¢lb<¢20. Hence Aut(EN*)~¢20.


Below we denote by L one of the formal systems EA, DA and by L* the corresponding semi-formal system.

Inductive definition of the autonomous ordinal terms of the semi-formal system L*. 1. If y is less than the limiting number of L then y is autonomous in L*. 2. If <5 is autonomous in L* and L* f2- J[y] holds where <5 < y, then y is also

autonomous in L*.

Lemma 4. If y is autonomous in L* then, for all n < ro, y + n is autonomous in L*.

Proof by induction on y. If y is less than the limiting number of L then y + n is also less than the limiting number of L and therefore y + n is autonomous in L*. Otherwise there is an autonomous ordinal term <5 <y such that

We may assume that ro 2 ~<5 and I ~n. By Lemma 7b) of §21 the formula

J[y] -+ J[n] -+ J[y + n]

is deducible in L. By Theorem 21.3 J[n] is also deducible in L. Using a cut and Lemma 3 of ~21 we obtain


L* f2-+1 J[y+n].

Here <5 + I < y + n and by I.H. <5 + I is autonomous in L*. Consequently y + n is also autonomous in L*.

Lemma 5. The autonomous ordinal terms of L* form a segment.

Proof This result follows in the same way as Lemma 3 since the formula

is deducible in L for yo<y.

Definition. Again following S. Feferman [1] we denote by Aut(L*) the least ordinal term which is not autonomous in L*.

Theorem 23.7. Aut(EA*)=¢20.

1. Proof of ¢20 ~ Aut(EA *). Let an be defined as in the proof of Theorem 23.6.


Then w2 + 2· O(n = O(n. Therefore from Theorem 21.4 we have

By Theorem 23.4 0(0 + 1 is autonomous in EA *. Using Lemma 4 it follows by induction on n that every ordinal term O(n is autonomous in EA *. As in the proof of Theorem 23.6 it follows that </120::::; Aut(EA *).

2. Proof of Aut(EA *)::::; </120. By Theorem 23.4 </120 is greater than the limiting number of EA. Suppose

EA* ~f[y]

where (j < </120. We have to prove that y is then also less than </120. By Theorem 22.13 we have

By two applications of Theorem 22.8 we obtain

By Theorem 23.2 we obtain y::::;w·</11(</11(w+(j»<</120. Hence </120 is not autonomous in EA * and therefore Aut(EA *)::::; </120.

Theorem 23.8. Aut(DA*)=ro.

1. Proof of ro::::;Aut(DA*). By Theorem 21.6 we have

DA* f£.+13 f['n+l].

By Theorem 23.5 '0 + 13 is autonomous in DA *. It follows by induction on n using Lemma 4 that every ordinal term 'n is autonomous in DA*. By Theorem 14.16 to each ordinal term y<ro there is a natural number n such that y<'n. Therefore by Lemma 5 every ordinal term < r 0 is autonomous in DA * and therefore r 0::::; Aut(DA *).

2. Proof of Aut(DA *)::::; r o. By Theorem 23.5 r 0 is greater than the limiting number of DA. Suppose

DA*~f[YJ

holds where (j < r o. We have to prove that y is then also less than r o. By Theorem 22.14 for 0": =wo we have


It follows by Theorem 22.7 that for 1X:=¢(w·a+w)(w·a+w+6·£5)<ro that

RA*lTI-f"U]'

By Theorem 23.2 we obtain y~w·lX<ro. Hence ro is not autonomous in DA* and therefore Aut(DA *) ~ r o.

4. The Autonomous Ordinal Terms of the System RA *

IfL is a formal system with the basic inference rule (ind) and L* is the semi-formal system obtained from L by replacing (ind) by (S2.0*) then by Theorem 20.13 infinite deductions in L* with orders < w 2 correspond to the finite deductions of the system L. It is therefore fitting to allow the following inductive definition of the autonomous terms of the system RA * to begin with the ordinal terms < w2 •

Inductive definition of the autonomous ordinal terms of the semi-formal system RA * : 1. Every ordinal term < w2 is autonomous in RA *. 2. If RA * I%- fl [YJ holds where (X and p are autonomous ordinal terms which

are less than y, then y is also autonomous in RA *.

Lemma 6. If y is autonomous in RA * and w2 ~ y, then y + 1 and wY are also autonomous in RA*.

Proofby induction on y. By the hypothesis there exist autonomous ordinal terms (X<y and p<y such that

It follows from Lemma 7 of §21 that

EA f- fey] -4 f[y+ I]

EA f- fey] -4 f[wY].

By Lemma 3 of §21 and Theorem 22.13 it follows that there exist £5 1 <W2 and £5 2 < w 2 such that

(2) RA* f*!~' fl[y] -4 fl[y+ I]

(3) RA* f*!~2fl[y] -4 fl[WY]'

We may assume that w+£5 1 ~IX, w+£52 ~(X and w+w~p. The formula fl[y] has a rank <w+w. Therefore by cuts from (1)-(3) we obtain

(4) RA*I%-+lfl[y+l]

(5) RA*I%-+l fl[WY]'


If IX<W2, then IX+ 1 <W2 is autonomous in RA*. Otherwise IX+ 1 is autonomous by the I.H. Since IX + 1 < y + 1, it follows by (4) that also y + 1 is autonomous in RA *. If wY = y, then wY is autonomous by the hypotheses. Otherwise y < wYand IX + 1 < wY•

Then wY is autonomous in RA * by (5).

Lemma 7. The autonomous ordfnal terms ofRA* form a segment.

Proof Suppose Yo<y and y is autonomous in RA*. We prove by induction on y that Yo is also autonomous in RA*. This is trivial for y~W2. Now suppose w2<y. Then there exist autonomous ordinal terms IX < y and p < y such that

For Yo<y we have

By Lemma 3 of §21 and Theorem 22.13 it follows that there exists ~<W2 such that

We may assume that W+~~IX and w+w~p. By a cut from (1) and (2) we obtain

By Lemma 6 IX + 1 is autonomous in RA *. If Yo ~ IX + 1 or Yo ~ P then Yo is autonomous in RA * by I.H., otherwise Yo is autonomous in RA * by (3).

Definition. As before we denote by Aut(RA *) the least ordinal term which is not autonomous in RA *.

Proof It follows from the hypothesis by Theorem 22.5 d) that

holds for every permissible predicator po of level o. The assertion follows using a basic inference (S2.1 *).

Theorem 23.9. Aut(RA *) = r o.


1. Proof that ro~Aut(RA*). By Theorem 21.3

EA f- ..F[(1]

where (I =4>10=80 .

It follows by Lemma 3 of §21 and Theorem 22.13 that there exists <5 < w 2 such that

Hence (I is autonomous in RA *. By Theorem 21.6 if n~ 1 then

DA* fb+13 ..F[(n + IJ.


where a:=w'n+ 13 . Now w·a+w+6·((n+13)<w'n+14. Therefore, we have by Lemma 8

RA* Iw' +14 a1[-r-] fW';:+14J Sn+l'

If C is autonomous in RA *, then by Lemma 6 C + 14 and w'n+ 14 are also autonomous in RA *. Since w'n + 14 < (n + l' it follows that (n + 1 is autonomous in RA *. By induction on n, every ordinal term (n(n ~ 1) is autonomous in RA *. Then by Theorem 14.15 and Lemma 7 every ordinal term <ro is autonomous in RA* and therefore r 0 ~ Aut(RA *).

2. Proof that Aut(RA*)~ro. Suppose

holds where rx < r 0 and p < r o. It follows by Theorem 22.7 that

Hence by Theorem 23.2

Consequently r 0 is not autonomous in RA * and therefore Aut(RA *) ~ r o.


5. The Limits of Predicativity

Predicative inferences are formalized in a natural way in a system of ramified analysis of the same sort as RA *. But such a system can only be seen to be predicative if the only ordinal terms permitted to be orders of deductions and levels of formulas are those for which transfinite induction can be proved using solely predicative means. If RA * ~ .f1 [y] holds then in RA * there is a deduction of the formula .f1 [YJ of order a in which only formulas of levelland levels < p occur. Therefore from the proof of Theorem 23.9 we see that: to every ordinal term a:;" w 2 and < rowe can fix a strictly increasing sequence of ordinal terms ao, ... , an such that ao = w 2 and an = a in such a way that a deduction of order < ai can be given for formalized transfinite induction up to ai + 1 (i < n) in which, besides formulas of levell, only formulas of level < ai occur. In this way transfinite induction is predicatively provable up to each ordinal term < r o' On the other hand, by Theorem 23.9, a deduction of transfinite induction up to r 0 must have an order :;" r 0 or it must contain occurrences of formulas of levels up to r o'

Therefore transfinite induction up to r 0 is not strirtly predicatively provable. Our formalization of transfinite induction is based on a given well-ordering

-< of the natural numbers. For a differently defined well-ordering Wthe associated ordinal number Autw(RA *) may be < r o' But the proof that Aut(RA *) ~ r 0 can be carried over, without any difficulty, to every well-ordering W. We also have:

1. For every ordinal term a < r 0 there is a predicatively provable well-ordering of order type a.

2. There is no predicatively provable well-ordering of any order type:;" r o. Thus r 0 is the limiting number for the predicative provability of transfinite

induction. (ef. S. Feferman [1]) The semi-formal system of A f -analysis with deduction of orders < r 0 must

now also be considered predicative since by Theorem 22.14 DA * can be interpreted in RA *. In DA * by Theorem 23.8 formalized transfinite induction is even provable autonomously up to the limits of predicative provability.

The proofs of Theorem 23.3~23.5 show that every formal system L we have considered can be embedded in a subsystem of RA * for which eliminability of cuts is provable by means of transfinite induction up to the limiting number of L. Hence we have a proof of the consistency of L by transfinite induction up to the limiting number for L. (Such a proof for pure number theory was given by G. Gentzen [2] and [3].) Likewise, for every semi-formal system L* (with deductions of orders < Aut(L*» we have considered we have a consistency proof by transfinite induction up to Aut(L*). These consistency proofs yield still stronger methods than for the demonstration of predicativity, since, for example, in order to deduce .f1[(n+ 1] in RA * we need a subsystem of RA * for which the elimination of cuts can be carried out only with the use of deductions of orders > (n + l'

Chapter IX

Higher Ordinals and Systems of ni-Analysis

Our aim in this chapter is the proof-theoretic treatment of lIt-Analysis following G. Takeuti [4]. In §§24-26 starting from normal functions we present an extended constructive system of ordinals with the necessary properties for this and in §29 we give a constructive proof of well ordering for this system. In §27 and §28 we carry out a constructive consistency proof for a formal and for a semiformal system of lIt-Analysis, using transfinite induction on the system or ordinal terms of §25.

§f4. Normal Functions on a Segment (])* of the Ordinals

In this section we consider ordinals in the sense of the classical theory of ordinals based on the axioms of Zermelo-Fraenkel or Bemays-GOdel set theory. These ordinals, as defined in §13, characterize the order types of the well-ordered sets. However, we do not deal here with the class of all ordinals but only with a proper segment 0* which nevertheless is larger than the segment 0 dealt with in §13. We characterize 0* itself by an axiom system without tying this to a general axiom system for set theory. Thus we use the notions map (function) and set in a naive way. But these may also be regarded as being determined axiomatically (in the context of a general axiom system for set theory).

1. Axiomatic Characterization of the Segment 0* of the Ordinals

Ax. I. 0* is a non-empty set well-ordered by the relation <. (See §13.1.)

We call the elements of 0* ordinals. We denote by 0 (zero) the least element of 0* (with respect to the relation <). In this section the letters rx, [3, y, 6, ~, 11, ( (possibly with indices) always denote ordinals from the set 0*. rx~[3 means that rx<[3 or rx=[3. rx> [3 (rx ?; [3) stands for [3<rx ([3~rx). We denote by O*(rx) the segment of all ordinals ~ < rx.

An ordinal rx is said to befinite (transfinite) if O*(rx) is a finite (infinite) set.

222 IX. Higher Ordinals and Systems of nl-Analysis

If M 1 and M z are subsets of 0* then M 1 '" M z denotes that there is a bijection of Ml onto M z. This relation is obviously an equivalence relation. An ordinal IX is said to be a cardinal if there is no ~ <IX such that O*@",I(])*(IX).

Ax. II. There is a map ~ from 0* to 0* with the following properties: 11.1. ~(IX) is a transfinite cardinal of 0*. 11.2. For each transfinite cardinal /3 of I(])* there is an IX such that ~(IX)=/3. 11.3. IfIX</3 and I(])*(~(IX»"'O*(~), then ~<~(/3). 11.4. IX<~(IX).

From 11.3 (for ~=~(IX» we have: If IX </3, then ~(IX)<~(/3). It follows by 11.1 and 11.2 that ~ is an order preserving map of I(])* onto the set of transfinite cardinals in 0*. In the classical theory of ordinals there is a map ~ with properties 11.1-11.3 from the class of all ordinals onto the class of all transfinite cardinals. This map has fixed points IX=~(IX). By 11.4 0* is restricted to the segment of those ordinals which are less than the least fixed point of~. .

By 11.4, for each ordinal IX there is an ordinal> IX in 0*. Using Ax. I it follows that for each IX there is a smallest ordinal >IX in I(])*. We call this the successor IX' of IX. An ordinal /3 is said to be a limit number if /3 '" 0 and there is no IX such that IX' =/3.

A set Me I(])* is said to be bounded if there exists /3 such that IX ~ fJ for all IX E M. By Ax. I for every bounded set Me I(]) * there is a unique least ordinal /3 such that IX ~ /3 for all IX E M. As previously we call this ordinal fJ the supremum sup M of the set M.

We say that a set M ~ 0* has cardinality IX if IX is a cardinal such that M", I(])*(IX). In this case IX is uniquely determined by M.

The closure of a set M 0 ~ 0* under finitely many ni-placed functions of ordinals I; (i = 1, ... , m) is the smallest set M ~ I(]) * such that M 0 ~ M and 1;(1X1' ... , IXn,) E M for alllX1, ... , IXn, E M (i= 1, ... , m).

Ax. III. If M is the closure of the set {O} u 0*(13) under finitely many functions of ordinals then every subset of M has cardinality ~max(/3, ~(O».

This property of 0* is provable in the classical theory of ordinals.

Ax. IV. If M is a subset of O*(~(IX'» which has cardinality < ~(IX') then sup M < ~(IX').

This property of 0* which we call the regularity of the cardinal ~(IX') is provable in the classical theory of ordinals using the axiom of choice.

Ax. I-Ax. IV comprise all the properties of the segment 0* of the ordinals which will be used in the sequel without reference to a general system of axioms for set theory.

2. Basic Properties of 0*

By Ax. I the induction theorem formulated in §13 (p. 75) also holds for 0*. As in § 13 we denote by N the set {O, 0', 0", ... } of finite ordinals. Then putting

w : = ~(O), w = sup N holds.

24. Normal Functions on a Segment 0* of the Ordinals 223

We define O*-segments and the ordering functions in the same way as in §13 (p. 77). Every proper O*-segment (that is, every O*-segment different from 0*) is a set O*(IX). Every set M~O* has precisely one ordering functionfwhich maps an O*-segment A onto M preserving order. Then IX ~f(lX) for all IX E A. We denote the domain A of the ordering functionfby D(f).

We also define continuity of an ordering function as in §13. A normal function on 0* is a continuous ordering function with domain 0*. A set M ~ 0* is said to be closed if for every nonempty bounded set M 0 ~ M we have sup MoE M.

Lemma l. ff the ordering function f of a set M ~ 0* has domain 0* and M is closed then f is a normal function.

Proof like that for Theorem 13.5.

We set w : = ~(O), Qo : = 0 and Qa : = ~(IX) for IX * O. Let K be the set of transfinite cardinals > w, then K: = {Qa IIX * O}.

Lemma 2. If K~M~O* then the ordering function of M has domain 0*.

Proof Letfbe the ordering function of M. It follows from K~M that for every ordinal IX E 0* there is an ordinal ~ E D(f) such that f(~) = Qa" where IX ~ ~. Thus IX E D(f) too. Hence D(f) = 0*.

Lemma 3. a) ljlX is transfinite, then O*(IX')~O*(IX). b) Every transfinite cardinal is a limit number.

Proof a) If IX is transfinite, then the function f given by fen) : = n' for n < w, f(~) :=~ for W~~<IX andf(lX) :=0 is a bijective map ofO*(IX') onto O*(IX).

b) follows from a).

Addition of ordinals is defined as in §13.5. Let Ba:={~IIX~~EO*},fa be the ordering function of Ba and 1X+{3:=fa({3) for {3ED(fa).

Lemma 4. If IX < {3 and {3 E K, then {3 E D(fa) and IX + {3 = {3.

Proof There exists Y E DUJ such that 0 < Y ~ IX + Y = {3. There are the following two cases to consider for {3 E K.

1. {3=~(c5'). If M:={IX+~ I ~<y} then M~O*(~(c5')) and M~O*(y). By Lemma 3b) ~(c5') is a limit number. Therefore sup M = ~(c5'). By Ax. IV it follows that M has cardinality ~ ~(c5'). Thus {3 = ~(c5') ~y. It follows that {3= y and therefore {3 E D(Ja) and IX + {3 = {3.

2. {3=~(c5) where c5 is a limit ordinal. Assume that y<{3. Then there exists ~ < c5 such that y < ~(n < {3 and IX < ~(~'). It follows from I. that ~(n E D(fa) and IX + Y < IX + ~(~') = ~(n < {3 contradicting IX + y = {3. Hence {3 = y and therefore {3 E D(fa) and IX + {3 = {3.

224 IX. Higher Ordinals and Systems of lI~-Analysis

Theorem 24.1. h. is a normalfunction.

Proof For each ordinal e E 0* there exists p E K such that rx < P and e < p. Then by Lemma 4 p E D(fa) and therefore e E D(h.) too. Hence D(h.) = 0*. Since Ba is closed it follows by Lemma 1 that the ordering function of Ba is a normal function.

By virtue of Theorem 24.1 we 0 btain all the properties of addition of ordinals developed in §13.5. We set 1 : =0',2: =0", .... Then rx' =rx+ l.

We define additive principal numbers and the ordering function rx f-+ wa as in §13. Corresponding to Theorem 13.7 there are, for each rx # 0, uniquely determined rxl~···~rxn such that rx=wal+···+wan. We call the additive principal numbers wal , ... , wan involved the components of the ordinal rx. The ordinal 0 has no components. We denote the set of additive principal numbers in 0* by P and the finite set of components of rx by P(rx). The set P(O) is empty. rx E P holds if, and only if, P(rx) = {rx}. In general we have pcP) ~P(rx+ P) ~P(rx) u PcP).

Theorem 24.2. K~P.

Proof If P E K then by Lemma 4 PEP.

Definition of the level Srx of an ordinal rx. Given rx there is a unique smallest ordinal P such that O*(P),.... O*(rx). This ordinal is a cardinal. If P < w let Srx: = O. Otherwise there exists y # 0 such that P = Qy. Then let Srx : = y.

Corollaries QSa ~ rx < Qsa. + 1 and SQy = y.

Proof If rx is finite then Srx=O and QSa.=O~rx<QSot+l' Otherwise there exists y such that O*(~(y)),....O*(rx). Then Srx=y, Qsa.~~(Y)~rx and by Ax. II.3

rx<~(y+ 1)=QSa.+l'

By definition we also have SQy=Y.

3. Definitions of the Functions () a.

Following a definition of S. Feferman and P. H. G. Aczel [I] we define functions ()a. as follows.

Inductive definition of sets of ordinals CnCrx, p), C(rx, p), In (rx) and ordering functions ()a:

(DI) Co(rx,p):=O*(P)

(D2) p(e)~ CnCrx, P) = e E Cn+ l(rx, P)

(D3) e<rx, e E C(e, ()e'1), '1<()e'1 and e, '1 E CnCrx, p) = ()e'1 E Cn+1(rx, P)

24. Normal Functions on a Segment I()I* of the Ordinals

(D4) (#0, (E Cia, f3) => Q~ E Cn+ lea, f3)

(D5) C(a, f3) : = U Cn(a, f3)

(D6) In (a): = {f31 f3 ¢ C(a, f3)} uK

(D7) Let 8a be the ordering function of the set In (a). We write 8af3 for 8a(f3).

225

This definition is inductive since it only assumes knowing the functions 8~ and sets C( (, 8(1]) for « a. Since K s; In (a) it follows by Lemma 2 that the ordering function 8a ofIn (a) has domain 0*. Hence 8af3 is defined for all a, f3 E 0*.

S. Feferman's definition refers to the closure Clif3) of the set {O} u 0*(f3) under addition and the functions 8~«(<a) and (f--+Q~ instead of C(a,f3). By W. Buchholz [l], C(a, f3)=Cla(f3). Thus the functions 8a defined here agree with the functions defined by S. Feferman. The condition ( E C((, 8(1]) in (D3) which is the essential difference from the definition of Clif3) is included since it makes the most important properties of the functions 8a easier to prove than on the basis of Clif3). The stratification of C(a, f3) into Cia, f3) (n E N) introduced by W. Buchholz helps in carrying out the proofs of induction.

The property f3 ¢ C(a, f3) occurring in (D6) says that f3 is inaccessible from ordinals <f3 by certain functions such that the only 8~ which occur among these functions are ones for which « a. We therefore denote by In (a) the set of ainaccessible ordinals. If a < r 0 then the set Cr (a) of a-critical numbers defined in §13 is a subset of In (a) as we shall show in §25.

4. Properties of the 8 Functions

Lemma 5. a) f3~8af3, b) f31 <f32 => 8af31 < 8af32' c) «f3=> (EC(a,f3), d) 0 E C(a, f3), e) 8af3 E P, f) yE Cn(a, f3) => P(y)s;Cia, f3), g) Cn(a, f3)s;Cn+ l (a, f3).

Proofs. a) and b) hold because 8a is an ordering function. c) holds by (DI) and (D5). d) holds by (D2) and (D5) since P(O) is empty. e) Assume that 8af3 ¢ P. Then y < 8af3 for all y E P(8af3) and therefore by (DI)

P(8af3) c CoCa, 8af3) and by (D2) and (D5) 8af3 E C(a, 8af3). By Theorem 24.2 if we also had 8af3 ¢ K, we should have 8af3 ¢ In (a) by (D6) contradicting (D7). Consequently 8af3 E P.

f) Suppose y E Cia, f3). If n =0 or YEP then trivially P(y) s; Cn(a, f3). If n # 0 and y ¢ P then from e) and Theorem 24.2 we have that y E Cn(a, f3) can only hold by (D2). In this case P(y)s;Cn- l (a,f3). It follows that for all YiEP(y), P(yJ= {yJ S; Cn - l (a, f3) and by (D2) Yi E Cia, f3). Thus P(y) s; Cn(a, f3).

226 IX. Higher Ordinals and Systems of IIi-Analysis

g) By f) from I' E Cn(a, fJ) we have P(y) S; Cn(a, fJ) and by (02) I' E Cn+ 1 (a, fJ). Hence Cn(a, fJ) S; Cn+ 1 (a, fJ)·

Theorem 24.3. a) I' E C(a, fJ) ¢;> P(y) S; C(a, fJ)· b) If ~ < ex, ~ E O(~, 8~11) and ~, 11 E C(a, fJ) then 8~11 E C(a, fJ)·

Proofs. a) By (05) and Lemma 5 f) we obtain P(y) S; C(a, fJ) from I' E C(a, fJ). If P(y) S; C(a, fJ) holds then by (05) and Lemma 5 g) there exists n such that P(y) S;

Cn(a, fJ). Hence by (02) and (05) I' E C(a, fJ)· b) If ~, 11 E C(a, fJ) then by (05) and Lemma 5 g) there exists n such that

~, YJ E Cn( a, fJ). If 11 < 8~11 then from ~ < a and ~ E C( ~, 8~11) by (03) and (05) we obtain 8~11 E C(a, fJ)· Otherwise 8~11 = 11 E C(a, fJ).

Lemma 6. a) a l ~a2' fJl ~fJ2 => C.(a l , fJI)S;Cn(a 2 , fJ2)' C(al , fJI)s;C(a2, fJ;J·

b) If a is a limit number, then C(a, fJ) = U C(~, fJ).

c) If fJ is a limit number, then C(a, fJ)= U C(a,l1). ~<P

Proofs. a) If ex l ~a2' fJl ~fJ2 then C.(al , fJI)s;Cn(a2, fJ2) by induction on n. By (05) we obtain C(al , fJ I) S; C(a2, fJ2)'

b) Suppose a is a limit number and I' E Cn(ex, fJ). We prove, by induction on n, that there exists ~ < a such that I' E C.( ~, fJ).

bl). Suppose I' E C.(a, fJ) by (01), then I' E Cn(O, fJ). b2). Suppose YEC.(a,fJ) by (02), then YiECn-l(ex,fJ) for all YiEP(y). By

LH. there exist ~ i < a such that Yi E Cn _ I (~i' fJ). Let ~ be the maximum of these ~ i then by a) P(y) S; Cn -I (~, fJ) and by (02) I' E C.( ~, fJ).

b3). If I' E C.(a, fJ) holds by (03) then 1'=81'11'2 where 1'1 <a and 1'1,1'2 E Cn-l(a,fJ). By LH. there exist ~i<a such that YiECn-IGi,fJ) (i=I,2). Let ~:= max(YI+l'~I'~2) then ~<a, YI<~ and by a) YI,Y2ECn-l(~,fJ). By (03) we obtain I' E C.(~, fJ)·

b4). Suppose I' E Cn(ex, fJ) holds by (04), then the assertion follows immediately from LH.

Hence Cn(a, fJ) S; U Cn(~' fJ) and consequently by (05) C(a, fJ) S; U C(~, fJ)·

By a) we also have U C(~, fJ) S; C(a, fJ)· ~<~

c) Suppose fJ is a limit number and I' E C.(a, fJ). We prove, by induction on n, that there exists 11 < fJ such that I' E Cn( a, 11). If n = 0, then this holds for 11 = I' + 1 since then I' + 1 < fJ. If n # ° then I' E C.( a, fJ) holds on the basis of finitely many hypothesis YiECn-l(a,fJ) (i=I, ... ,m). By I.H. there exist l1i<fJ such that YiE Cn-l(a, YJJ Let 11 : = max (YJI' ... , 11m) then by a) Yi E Cn- l (a, 11) (i = 1, ... , m). Hence I' E Cia, 11)· Thus Cn(a, fJ)c U Cn(a,l1)· By (05) we obtain C(ex, fJ)s; U C(a,l1)·

~<P ~<P

By a) we also have U C(a, YJ)s;C(a, fJ). ~<P

24. Normal Functions on a Segment O· of the Ordinals 227

Theorem 24.4. a) In (O)=P, ()Of3=oI b) ex l <ex2 ~ In (ex2) sIn (ex l ), ()exlf3~()ex2f3.

c) If ex is a limit number, then In (ex) = n In (e). ~<"

Proofs. a) If f3 E P and f3 ¢ K then we obtain f3 ¢ Cn(O, f3) by induction on n. By (D5), (D6) and Theorem 24.2 we obtain PsIn (0). By Lemma 5 e) we also have In (0) S P. Hence In (0) = P. It follows that ()0f3 = wP, since f3 f-+ wP is the ordering function of P.

b) If ex l <ex2 then by Lemma 6 a) C(ex l ,f3)SC(ex2 ,f3). By (D6) we obtain In (ex2) S In (ex l ) and by (D7) ()ex l f3 ~ ()ex2f3 by induction on f3.

c) Follows from Lemma 6 b) by (D6).

Lemma 7. For each ex, f3 there exists a least ordinal y ¢ C(ex, f3). This ordinal y has the properties: f3 ~ y, Sf3 = Sy, C(ex, f3) = C(ex, y) and y E In (ex).

Proof. Let M be the closure of the set {O} u O*(f3) under addition and the functions (e, 1]) f-+ ()e1] and e f-+ Q~. By (Dl)-(D6) C(ex, f3)sM and therefore we also have Mo:=C(ex, f3) n O*(QSP+l) sM. By Ax. III Mo has cardinality~max (f3, ~(O»<QSP+l· Hence Mo is a proper subset of O*(QSP+l). Therefore there exists a least ordinal y<QSP+l such that y¢ C(ex, f3). By Lemma 5 c) f3~y.

Since y < QSfJ + l' Sf3 = Sy. By the minimality of y ¢ C( ex, f3) we have e E C( ex, f3) for aU e<y. Consequently Co(ex,y)sC(ex,f3). It follows by induction on n that Cn(ex, y)s C(ex, f3) for all n E ~. Therefore by (D5) C(ex, y)s C(ex, f3). By Lemma 6 a) we also have C(ex, f3) S C(ex, y). Thus y ¢ C(ex, f3) = C(ex, y) and by (D6) y E In (ex).

Lemma 8. a) C(ex, 0) = C(ex, ()exO). b) C(ex, ()exf3 + 1) = C(ex, ()ex(f3 + 1».

Proofs. a) By Lemma 7 there exists y E In (ex) such that C(ex, 0) = C(ex, y). It follows that ()exO ~ y and by Lemma 6 a)

C( ex, 0) S C( ex, ()exO) S C( ex, y) = C( ex, 0),

and therefore C( ex, 0) = C( ex, ()exO). b) By Lemma 7 there exists y E In (ex) such that C(ex, ()exf3 + 1) = C(ex, y) and

()exf3 + 1 ~ y. It follows that ()exf3 +- 1 ~ ()ex(f3 + 1) ~y and by Lemma 6 a)

C(ex, ()exf3 + 1) s C(ex, ()ex(f3 + 1» s C(ex, y) = C(ex, ()exf3 + 1),

and therefore C( ex, ()exf3 + 1) = C( ex, ()ex(f3 + 1».

Theorem 24.5. a) S()exf3 = Sf3. b) ()exy = y for y E K.

228 IX. Higher Ordinals and Systems of n:-Analysis

Proofs. a) There exists y such that QS/J+l=Orxy since QSP+1EKS;In(rx). For ~<QSP+l' by Lemma 7, there exists '1EIn(rx) such that ~~'1<QSP+l' Let M:= {Orx'1l '1 <y} then sup M = Qsp+ l' Thus Ms; O*(Qsp+ 1) and M", O*(y). By Ax. IV we obtain QSP+l ~y. Therefore

Hence SOrxf3 = Sf3. b) If y E K there exists 13 such that y = Orxf3 since K S; In (rx). By a), Sy = S 13. It

follows that

and therefore f3=y and Orxy=y.

Theorem 24.6. a) Qy E C(rx, 13) ~ y E C(rx, 13)· b) y E C(rx, 13) = QSy E C(rx, 13)·

Proofs., a) The assertion is trivial if y=O. Now suppose y#O. If Qy E C(rx, 13) then there is a least number n such that Qy E CnC rx, 13). If n = 0 then y < Qy < 13, and therefore, also, y E Co(rx, 13). If n #0, then by Theorems 24.2 and 24.5 b) we have that Qy E C(rx, 13) can only hold because of (D4). Therefore y E Cn- l (rx, 13). Therefore from Qy E C(rx, 13) we have, in every case, y E C(rx, 13). By (D4) and (DS) from y E C(rx, 13) we obtain Qy E C(rx, 13).

b) Suppose y E C(rx, 13) holds then there exists n such that y E CnCrx, 13). By Theorem 24.5 a) we obtain QSy E C(rx, f3) by induction on n.

Proof Suppose y E Cn(rx2, 13). We prove by induction on n that y E Cn(rxl , 13). If y E Cn(rx2, 13) holds by (DI), (D2) or (D4), then y E Cn(rx l , 13) follows either trivially

. or by the I.H. Now suppose YECn(rx2,f3) holds by (D3). Then y=O~'1 where ~ < rx2 and ~, '1 E Cn- l (rx2, 13)· By I.H. we have ~, '1 E Cn- 1 (rx l , fJ). Using ~ < rx2 and the hypothesis \f~(rxl ~~<rx2 ~ ~ ¢ C(rx l , 13» we obtain ~<rxl' By (D3) we obtain YECnCrx l ,f3). Hence CnCrxz,f3)S;CnCrxl,f3) holds. By (DS) we obtain C(rx2,f3)s; C(rx l ,f3)· By Lemma 6 a) C(rxl , 13) S; C(rx2, 13) also holds.

Theorem 24.7. a) rx l < rx2, \f~(rxl ~ ~ < rx2 ~ ~ ¢ C(rxl , Orx2f3» = Orxl f3 = Orx2f3. b) 3~(rxl ~ ~ < rx2 J\ ~ E C( rx l , Orx2f3» = Orx l (Orx2f3) = Orx2f3.

Proofs. a) When '1 ~Orx2f3 we obtain

from the hypothesis. By Lemma 6 a) and by Lemma 9 C(rx l , '1)=C(rx2, '1). By (D7) we obtain Orx l f3 = Orx2f3.

24. Normal Functions on a Segment iIJI* of the Ordinals 229

b) If erJ,2{3 E K, then erJ,l (erJ, 2 {3) = erJ,2{3 holds by Theorem 24.5 b). Now suppose erJ,2{3 rj K. By Theorem 24.4 b) rJ,1 :( ~ < rJ,2 implies erJ,2{3 E In m. Therefore there exists '1:(e~'1=erJ,2{3. Using ~ E qrJ,1,@rJ,2{3), by Lemma 6 a) we obtain

If we had '1 < erJ,2{3, then by Lemma 5 c) we should have '1 E qrJ,2' erJ.2{3) and by Theorem 24.3 b) e~'1 E q rJ,2' erJ,2{3) which contradicts (D6) since e~1J = erJ,2{3 rj K. Thus '1=erJ,2{3. By Theorem 24.4 b) we have

and therefore erJ,l (erJ,2{3) = erJ,2{3.

Lemma 10. a) IferJ,l{3l =erJ,2{32 where rJ,1 :(1X2 , then PI ={32 or {3l =erJ,2{32' b) rJ,1 <rJ,2, rJ,1 E qrJ,2' erJ,2P) = erJ,1(erJ,2{3) = eIX2{3·

Proofs. a) If 1X1 :(rJ,2 then by Theorem 24.7 erJ,1{32 =erJ,2~ or erJ,1(ea2{32)=erJ,2{32' IferJ,l{3l =erJ,2{32' then {3l ={32 or {3l =erJ,2{32'

b) Suppose rJ,1 < rJ,2 and al E qrJ,2' erJ,2{3). IflXl rj qrJ,l' ea2/3) then qrJ,l' erJ,2{3) # qrJ,2, erJ,2{3)· Then, by Lemma 9, there exists ~ E qrJ,l' erJ,z/3) such that rJ,1 ~~<rJ,2' By Theorem 24.7 b) it also follows in the case that a l E qal, erJ,2{3) that erJ,l (ert.2/3) = ed.2 /3·

Lemma 11. erJ,{3 rj K, erJ,{3:( l' E q rJ" erJ,{3) = S {3 < Sy.

Proof By hypothesis there exists m such that l' E Cm(rJ" erJ,{3). We prove S{3 < Sy by induction on m. Since erJ,/3:(y, m#O therefore m=n+ 1.

1. Suppose l' E Cn+ 1(rJ" erJ,/3) holds by CD2). Since erJ,/3 E P, it follows from erJ,{3:(y that there exists Yo E P(y) where ea/3:(yo' Hence Yo E Cn(cx, erJ,/3). By I.H. we obtain S{3 < SYo' Therefore we also have S{3 < Sy.

2. Suppose l' E Cn + 1 (a, erJ,{3) holds by (D3). Then l' = e~'1 where ~ < rJ, and ~, '1 E CnCrJ" erJ,/3). By Lemma 10 b) we have ()~(ea{3) = erJ,{3. Using erJ,{3:( l' = e~'1 we obtain erJ,{3:('1. By I.H. we obtain S{3<S'1. Then by Theorem 24.5 a) we also have S{3 < Sy.

3. Suppose l' E Cn+ 1(rJ" erJ,{3) holds by (D4). Then l' E K. Since erJ,{3 rj K and erJ,{3:(y, we have ecx/3<y and S{3<Sy.

Theorem 24.S. a) {3 rj K = qa, ea{3) n O*(Qsp+ 1) = o*(erJ,{3). b) qrJ" 0) n O*(Ql) = o*(erJ,O).

Proofs. a) By Lemma 5 c) we have

(1) o*(erJ,{3)r;;qrJ" erJ,/3).

230 IX. Higher Ordinals and Systems of l1~-Analysis

By Theorem 24.5 a) ()a.f3< QS/i+ 1 and therefore

(2) 0*«()a.f3) S O*(QS/i + 1)·

By Theorem 24.5 b) f3 ¢ K implies ()a.f3 ¢ K. If ()a.f3:::; y E C( a., ()a.f3) then by Lemma 11 QS/i+l :::;y. Hence we also have

Assertion a) follows from (l}-{3). b) Follows from a) since SO=O ¢ K and by Lemma 8 a) C(a., 0) = C(a., ()a.0).

Theorem 24.9. ()" is a normal function.

Proof Since ()" has domain 0*, by Lemma 1 it remains only to prove that· In (a.) is closed. Let M be a non-empty bounded subset of In (a.). We have to prove that sup M is then a member of In (a.). If sup ME K or sup ME M then trivially sup ME In (a.). Now suppose sup M ¢ K and sup M ¢ M. Then sup M is a limit number. Assume that sup ME C(a., sup M). Then by Lemma 6 c) there exists ", < sup M such that sup ME C( a., ",). Since M S In (a.) and sup M ¢ K there is then ()a.e ¢ K such that ",:::; ()a.e :::; sup M and S()a.e = Se = S sup M. By Lemma 6 a) sup ME C(a.,,,,) implies sup ME C(a., ()a.e). By Lemma 11 we have Se < S sup M contradicting the choice of e. Consequently sup M ¢ C(a., sup M) and by (D6) sup ME In (a.).

-5. The Sets In(a.) and Functions ()"

We introduce sets In (a.) of ordinals and functions ()" on the ordinals following W. Buchholz [2] and use these to develop a notation system for a segment of the ordinals.

Definition. In (a.) : = In ( a.)\In (a. + 1). That is to say y E In (a.) is to hold if, and only if, y E In (a.) and y ¢ In (a. + 1).

In this case by Theorem 24.4 b) YEIn(e) for all e:::;a. and y¢In(",) for all

", > a.. In (a.) is therefore the set of those ordinals which are maximally a.-inaccessible.

Corollaries. 1. In ( a.) S P.

2. In (a.) and K are disjoint sets. - -

3. If a. l #a.2 then In (a. l ) and In (a. 2 ) are diSjoint sets.

Theorem 24.10. In (a.) is the set of those ordinals ()a.f3 such that a. E C(a., ()a.f3) and {3 < ()a.f3.

24. Normal Functions on a Segment (]I* of the Ordinals 231

Proof l. Suppose Y E In (a), then Y E In (a) and y ¢ In (a + 1). Then there exists f3 such that y = ()af3 ¢ K, ()af3 ¢ C( a, ()af3) and ()af3 E C( a + 1, ()af3). It follows that C(a, ()af3)"# C(a+ 1, ()af3) and by Lemma 9 a E C(a, ()af3). Since ()af3 E C(a + 1, ()af3) there exists a least number n such that ()af3 E Cn<a+ 1, ()af3). Because ()af3 E P and ()af3 ¢ K this can only hold by (D3). Then we have 1] < ()~1] = ()af3 where ~::( a. By Theorem 24.4 b) we have ()af3::( ()a1]. Then f3::( 1], and therefore f3 < ()af3.

2. Suppose a E C(a, ()af3) and f3 < ()af3. By Theorem 24.5 b) we then have ()af3¢K. By Lemmata 6 a) and 5 c) it follows that aEC(a+1,()af3) and f3E C(a+l,()af3). By Theorem 24.3 b) we obtain ()af3E C(a+l, ()af3). By (D6) using

()af3 ¢ K we have ()af3 ¢ In (a + 1). Hence we have ()af3 E In (a).

Lemma 12. ~ E C(~, ()~1]), 1] < ()~1] and f3::( ()~1] E C(a, f3) imply ~ < aand~, 1] E C(a, f3).

Proof By Theorem 24.5 b) 1] < ()~1] implies ()~1] ¢ K. Since ()~1] E C( a, f3) there is a least number n such that ()~1] E Cn< a, f3). Since f3::( ()~1] E P and ()~1] ¢ K this can only hold by (D3). Then we have ()~1]=()~01]0 wherel.o<a, ~oEC(~o,()~o1]o),

1]0 < ()~01]0 and ~o, 1]0 E C(a, f3). By Theorem 24.10 ()~1] E In m and ()~01]0 E In (~o)·

If ~"#~o, In (0 and In(~o) are disjoint. Therefore from ()~1]=()~01]0 we have ~ = ~o. So we also have 1] =1]0 and therefore ~ <a and ~, 1] E C(a, f3).

Definition. We set f1(a) to be the least ordinal 1] such that a E C(a, ()a1]). (Such an ordinal f1(a)::( a + 1 exists since a E C(a, ()a(a + 1 )).

Lemma 13. f1(a) is not a limit ordinal.

Proof If f1(a) were a limit number, then we should have

since by Theorem 24.9 ()a is continuous. Then, by Lemma 6 from a E C(a, ()a(f1(a))) we should have 1] <f1(a) such that a E C(a, ()a1]). This contradicts the minimality of f1(a).

Lemma 14. (E Cn(a, f3), ( ¢ C(y, ()yc5), (E C(y, ()yc5 + 1) => c5 E C(a, f3).

Proof By induction on n. By Lemma 5 c) (¢ C(y, ()yc5) implies ()yc5::«(. 1. Suppose ( E Cn< a, f3) holds by (D 1). Then ( < f3. It follows that c5::( ()yc5::( ( < f3

and c5 E C(a, f3). 2. Suppose (E Cn(a, f3) holds by (D2). By Theorem 24.3 a) (¢ C(y, ()yc5)

implies that there exist (0 E P«) such that (0 ¢ C(y, ()yc5). Hence (0 E Cn-1(a, f3) and (0 E C(y, ()yc5+ 1). By I.H. we obtain c5 E C(a, f3).

3. Suppose (ECn<a,f3) holds by (D3). Then we have (=()~1] where ~<a, ~ E C(~, ()~1]), 1] < ()~1] and ~, 1] E Cn - 1 (a, f3).

3.1. Suppose ()yc5 = (. By Theorem 24.10 ()~1] E In (~). Therefore from ()yc5 =()~1] we obtain y~~ and by Lemma 10 a) c5=1] or c5=()~1]=y; therefore in every case c5 E C(a, f3).

232 IX. Higher Ordinals and Systems of In-Analysis

3.2. Suppose ()yb«. Then ()yb+ 1 ';;;;()~11 E C(y, ()yJ+ 1). By Lemma 12 we obtain ~<y and~, 11 E C(y, ()yJ+ 1). Then by Theorem 24.3 b) from ()~11 ¢ C(y, ()yJ) we obtain ~ ¢ C(y, ()yJ) or 11 ¢ C(y, ()yJ). By LH. (for ~ or 11 replacing 0 we obtain J E C(a, [3).

4. Suppose (E Cn(a, [3) holds by (D4). Then we have (= Q~ where ~ECn-l(a,[3). By Theorem 24.6 a) ~¢C(y,()yJ) and ~EC(y,()yJ+l) also hold. By LH. we obtain J E C(a, [3).

Lemma 15. y E C(a, [3) ~ J1(Y) E C(a, [3).

Proof If J1(Y) = 0, then J1(Y) E C(a, [3) holds by Lemma 5 d). Otherwise by Lemma 13 J1(y)=J+ 1 where y¢ C(y, ()yb) and YE C(y, ()y(J+ 1)). By Lemma 8 we obtain y E C(y, ()yJ + I). Using y E C(a, f3) by Lemma 14 we obtain J E C(a, [3). By Lemma 6 a) from y E C(a, [3) and y ¢ C(y, ()yJ) we obtain y<a or ()yJ<[3 and therefore a #0 or 1 < [3. If a #0 then, since 0 E C(a, [3) by Theorem 24.3 b) 1 = ()OO E C,(a, [3). If 1 < [3 then 1 E C(a, [3) by Lemma 5 c). Therefore in every case J E C(a, [3) and 1 E C(a, [3). By Theorem 24.3 a) we obtain J1(Y) = J + 1 E C(a, [3).

Definitions. 1. If SJ1(a),;;;; S[3, set ri[3) to be the unique ordinal determined by [3 = QSI'(a) + ra([3)·

2. If [3 = [30 + n where n < wand J1( a) < [30 = ()a[3o, then set la[3 : = 1. Otherwise set la[3 : = O.

3. If SJ1(a),;;;; S[3 set (}a([3) : = ()a(J1(a) + ri[3) + laf3). We write Oa[3 for (}i[3).

Theorem 24.11. Ba is an order preserving map of the set of ~ ~ QSI'(a) onto the set

In (a).

Proof 1. Suppose [3~QS/i(al' then SJ1(a),;;;;S[3. We prove that Oa[3 E In (a). We have (}a[3 = ()ap where

P = J1(a) + ri[3) + la[3.

By Lemma 6 a) from aE C(a, ()C((J1(a))) and ()a(J1(a)),;;;;()ap we have

(1) aEC(a,()ap).

Assume that ()ap = p. Then 1 < PEP, consequently la[3 = 0 and by Lemma 13 ra([3) # O. Hence

13 = J1(a) + ri[3) = ri[3)·

Using QSI'(a) ';;;;J1(a) we obtain [3= Qs*) + ri[3),;;;;/3 =ra([3) ';;;;[3,

and therefore [3=P, ()a[3=[3 and J1(a) <[3. But in this case la[3= 1. Hence ()ap#p and therefore

(2) P < ()ap.

By Theorem 24.10 from (1) and (2) we obtain Oa[3=()ap E In (a).

24. Normal Functions on a Segment O· of the Ordinals 233

2. Suppose Y E In (ex). We have to prove that there exists f3 such that Stt(ex) ~Sf3 and Y = {}o:f3. By Theorem 24.10 there exists 11 such that y = ()ex/3, ex E C( ex, ()ex/3) and 11<()ex/3. By the definition of tt(ex) and Lemma 6 a) we obtain ()ex(tt(ex))~()ex11 and therefore tt(ex)~/3. Therefore there exists f3* such that

11 = tt(ex) + f3*.

If f3*=f3o+n where tt(ex)<f3o=()exf3o and n<w then it follows from /3<()ex/3 that n>O. In this case let f3:= Qsl1(IZ)+f3o+(n-l). Otherwise let f3:= Qsl1(IZ)+f3*. In any case we then have Stt(ex)~Sf3 and

/3 = tt(ex) + ra(fl) + lexf3,

and therefore y = ()ex/3 = {}exf3. 3. Suppose QSI1(IZ)~f31 <f32· We prove that {}exf31 < {}exf32· We have {}exf3;=

()ex11; where

Q~I1(a)~f31<f32 implies rif31) <ra(f32)' Iflexf31=0 then 111</32' If lexf31=1 and rif31) + I =rif32) then we also have lexf32=1. In every case /31 </32' Therefore {}ex 1f31 < (}ex 2f32'

Theorem 24.12. If Sttm ~ SYf then a) f3~{}~Yf E C(ex, f3) = ~ <ex and~, Yf E C(ex, f3). b) ~ < ex and ~, Yf E C(ex, f3) = (}~Yf E C(ex, f3).

Proof We have (}~Yf=()~i1 where i1=tt(~)+r~(Yf)+I~Yf. By Theorems 24.10 and 24.11 ~ E C(~, ()~i1) and i1<()~i1.

a) Suppose f3~ ()~Yf E C(ex, f3). By Lemma 12 we have ~ < ex and ~, i1 E C(ex, f3). Hence by Lemma 15 and Theorem 24.6 b) we have QSI1(~) E C(ex, f3) and by Theorem 24.3 a) r~(Yf) E C(ex, f3). Then by Theorem 24.3 a)

also holds. b) Suppose ~ <ex and~, Yf E C(ex, f3). Hence by Lemma 15 we have tt(~) E C(ex, f3)

and by Theorem 24.3 a) r~(Yf) E C(ex, f3). Since ex#O, 1 E C(ex, f3). By Theorem 24.3 a) we obtain

ii = tt(~) + r~(Yf) + lexf3 E C(ex, f3).

Using ~ < ex, ~ E C(~, ()~ij) and ~ E C(ex, f3), by Theorem 24.3 b) we obtain {}~Yf = ()~i1 E C( ex, f3).

Theorem 24.13. If Stt(ex) ~ Sf3 then ex E C(ex, (}exf3), f3 < {}exf3 and S{}exf3 = Sf3.

234 IX. Higher Ordinals and Systems of ill-Analysis

Proof We have 8ap = 8a/3 where /3 = f1.(a) + riP) + lap. By Theorems 24.10 and 24.11, a E C(a, 8a/3) and /3 < 8a/3. It follows that a E C(a, 8ap) and

P = QS/l(IZ) + r iP) ~ P < 8ap.

Sf1.(a)~Sp implies S/3=Sp. Using Theorem 24.5 a) we obtain S8ap=Sp.

Theorem 24.14. If Sf1.(aJ ~ SPi (i= 1,2), then a) 8a l PI =8a2P2 => a l =a2, PI =P2· b) a l <a2, a l E C(a2, 8a2P2)' PI <8a2P2 => 8al PI <8a2P2.

Proof a) By Theorem 24.11 8aiPi E In (aJ If al #-a2 then In (a l ) and In (a2) are disjoint. Therefore from 8a l Pl =8a2P2 we obtain a l =a2. Then by Theorem 24.11 we also have PI = P2.

b) It follows from the hypotheses by Lemma 2 c) that PI E C(a2, 8a2Pi) and by Theorem 24.12 b) that 8a l Pl E C(a2, 8a2P2). By Theorem 24.13 PI <8a2P2 implies 8a l P I < QSP2 + 1· Since 8a2P2 ¢ Kby Theorem 24.8 a) we have 8a lPI < 8a2P2·

§25. A Notation System for Ordinals Based on the ()a. Functions

1. The Set 8(Q) of Ordinals

Inductive Definition of the set 8(Q) c (D* and the degree Ga E N of an ordinal a E 8(Q):

1. 0 E 8(Q), GO: =0. 2. O<y ¢ P, P(y)c8(Q) => Y E 8(Q), Gy: = max {GYi I Yi E P(y)} + 1. 3. a, P E 8(Q), Sf1.(a)~Sp => 8ap E 8(Q), G8ap: = max (Ga, GP)+ 1. 4. O<a E 8(Q) => QIZ E8(Q), GQIZ : =Ga+ 1.

Obviously, for every ordinal a such that a E 8(Q) at most one of the above four defining rules can hold. It follows by Theorem 24.14 a) that every ordinal a E 8(Q) has a uniquely determined degree Ga EN.

2. Sets of Coefficients

We introduce finite sets of coefficients for ordinals Y E 8(Q) in such a way that Y E C(a, 8ap) can be characterised in a useful recursive way.

Inductive Definition of the coefficient sets Kv*Y and KvY for v E (D* and Y E 8(Q):

1. Kv*0=KvO:=0(emptyset).

2. O<y ¢ P => Kv*Y: = U KV*Yi' KvY: = U KvYi· YiEP(y) YiEP(y)

25. A Notation System of Ordinals Based on the 6. Functions 235

3. Y E]>, Sy~v => K:y: = 0, K,Y: = {y}. 4. y=8e'1, v<Sy => K:y :={e} u K,*e u K,*'1, K,y :=K,e uK,Yf. 5. y=Q~, v<e=Sy => K,*y :=K,*e, K,y :=K,e.

Corollaries. 1. Kv*y is a finite set of ordinals of 8(Q) which is empty if Sy ~ v. 2. K,y is a finite set of principal numbers for addition from 8(Q) whose levels

are ~v.

Notations for M c: {}(Q) and rx E 0* : 1. M < rx means that e < rx for all e E M. (If M is empty then M < rx for all

rxEO*.) 2. rx ~ M means that there exists e E M such that rx ~ e.

Theorem 25.1. /f 13 ¢ K and y E 8(Q) then y E C(rx, (}rxf3) holds if, and only if, Ks~y < rx and Kspy < (}rxf3.

Proof By induction on Gy. 1. y ¢ P. Then the assertion follows from the I.H. by Theorem 24.3 a). 2. YEP where Sy ~ Sf3. Then Ktpy =0< rx. In this case by Theorem 24.8 a)

y E C(rx, (}rxf3) holds if, and only if, y < (}rxf3 and therefore Kspy = {y} < (}rxf3. 3. y=8eYf where Sf3<Sy. Then (}rxf3<8eYf. In this case by Theorem 24.12 a)

y E C( rx, (}rxf3) holds if, and only if, e < rx and e, Yf E C( rx, (}rxf3). By I.H. this is the case if, and only if, Ktpy={e} u Ktpe u KtpYf<rx and Kspy=Kspe u KspYf<(}rxf3.

4. y=Q~ where Sf3<e=Sy. Then the assertion follows from the I.H. by Theorem 24.6 a).

Corollary to Theorem 25.1. /f S JL( a) ~ S 13 and y E 8( Q) then y E C( a, 8af3) if, and only if, Ktpy<rx and Kspy < 8af3.

Proof There exists /1 ¢ K such that {}af3 = (}a/1 and Sf3 = S/1. Therefore the assertion holds by Theorem 25.1.

Theorem 25.2. If (}af3 E {}( Q) then Ktprx < rx and Kspa u {f3} < 8rxf3.

Proof By Theorem 24.13 we have a E C(a, (}af3) and 13 < (}af3. Hence the assertion follows by the corollary to Theorem 25.1.

Theorem 25.3. /fa, 13 E 8(Q) then SJL(a)~ Sf3 if, and only if, Ktpa<a.

Proof 1. Suppose SJL(a) ~ Sf3. Then {}af3 E {}(Q). By Theorem 25.2 we obtain Ktpa<a.

2. Suppose Ktprx < a. Let y be the maximum element of the finite set {Qsp} u Ksprx, then Kspa<y+ 1 ~(}a(y+ 1), Sf3= S(y + 1) and y+ 1 ¢ K. By Theorem 25.1 we obtain a E C(rx, (}a(y + 1». By the definition of JL(a) it follows that JL(a) ~ y + 1. Therefore SJL(rx) ~ Sf3.


Theorem 25.4. If eIXJ3i E e( Q) (i = 1, 2) where S 131 = S 13 2 = v then erx 1 131 < eIX2f3 2 if, and only if, one of the following three conditions holds:

a) 1X1 < !Xl' KvlXl U {f3d < e1X2f32' b) 1X1 = 1X2 , 131 < 132' c) 1X2 <lXI' (j('J.lf31 ~Kv1X2 U {f32}'

Proof Suppose condition a) holds. By Theorem 25.2 Kv*lXl < 1X1 < 1X2 . Using KvlXl < fJIY. 2f32' by the corollary to Theorem 25.1, we obtain 1X1 E C(1X2 , fJIX2f32)' Using 1X1 <!X2 and 131 <fJrx2f32' by Theorem 24.14 b) we obtain ea l f31 <fJrx2 f32.

2. Suppose condition b) holds. Then tJrx l f31 < (}rx2f32 holds by Theorem 24.11. 3. Suppose condition c) holds. Then, by Theorem 25.2, {}1X 1f3 1 ~Kv1X2 U

{f32} < {}rx 2 f32 and therefore erx l f31 < {}1Y. 2 f32· 4. Suppose none of the conditions holds. Then either tJrx 1f31 = (}rx.2f32 or by

a)-c) (with SUbscripts interchanged) eIX2f32 < {}a l f31'

3. The Systems T* and OT* of Terms

Inductive Definition of the system of terms T* and of the degree Gc E N of a term cET*.

1. OET*, GO:=O. 2. If ci is a term eaibi or Qa, in T* for i = 1, ... , n (n ~ 2) then set c 1 + ... + cn E

T* and G(cl + .. , +cn): =max (Gc l , ... , GCn) + 1. 3. (I, bET* => eabET*, Geab :=max(Ga, Gb)+ 1. 4. O#a E T* => Qa E T*, GQa: =Ga+ 1. (Here O#a means that 0 and a are

terms of different forms). We call the terms in T* of the form 'eab or Qa' principal terms.

Inductive Definition of the notation Y E T* for an ordinal Y E e(Q): 1. 0:=0. 2. IfY=Yl + .. , +Yn E e(Q) where Yl ~ ... ~Yn (n~2) are principal numbers for

addition, then set Y : =Yl + ... + Yn' 3. If Y = elXI3 E fJ(Q), then set Y : = eEifJ. 4. If y = Qa E e(Q) where IX # 0, then set Y : = Q~. This is an inductive definition on the degree Gy of the ordinal Y E e(Q).

Obviously the term Y E T* has the same degree as the ordinal y.

The notation system OT*

Let OT* be the subsystem of T* which contains only ordinal terms Y E T* which denote ordinals Y E e(Q). This system OT* of terms is a system of unique notations for the set of ordinals fJ(Q). That is to say: Each term Y E OT* denotes one and only one, ordinal Y E e(Q) and each ordinal y E tJ(Q) is denoted by one, and only one term Y E e(Q). The remaining terms in T* which do not belong to OT* are formal strings of symbols which do not denote ordinals.

25. A Notation System of Ordinals Based on the 9. Functions 237

We set OT* = lJ(Q) whereby we identify each ordinal Y E lJ(Q) with its notation Y E OT*. Also we now call the ordinals of lJ(Q) ordinal terms (because of the uniqueness of the notations).

We say that two ordinal terms Yl, Y2 E lJ(Q) are recursively comparable if it can be decided whether Yl <Y2, Yl =Y2 or Y2 <Yl'

It is obviously decidable whether any string of symbols is a term ofT*. Below we prove that it is also decidable for every term c E T* whether c is an ordinal term of the system lJ(Q) and that two ordinal terms from lJ(Q) are recursively comparable.

If MclJ(Q) then GM <n (GM~n) means that G~<n (G~~n) for all ~ E M.

Lemma 1. a) 0 < oc E lJ(Q) => Soc E lJ(Q), GSoc < Goc. b) v E 0*, oc E lJ(Q) => GKv*oc< Goc, GKvoc~ Goc.

Proof By induction on oc.

Theorem 25.5. a) It is decidable,for c E T*, whether C E lJ(Q). b) Any two ordinal terms Yl, Y2 E lJ(Q) are recursively comparable.

Proof of a) and b) by induction on the maximum of 2Gc + 1 and 2Gy, + 2Gn. a 1) Suppose C is O. Then C E lJ(Q). a 2) Suppose C is c 1 + ... + Cn (n;;:,: 2) with principal terms C 1, ••• , Cn' By I.H.

it' is decidable, for i = 1, ... , n, whether Ci E lJ(Q). If Ci ¢: lJ(Q), then c ¢: lJ(Q). Now suppose c1, ••• ,CnElJ(Q). 2Gc'+2GC,+'<2Gc+l for i=l, ... ,n-l and therefore by I.H. ci is recursively comparable with ci+ l' If C1 ;;:,: .•• ;;:,:cn, then C E lJ(Q). Otherwise C ¢: lJ(Q).

a 3) Suppose C is lJab. By I.H. it is decidable whether a and b belong to lJ(Q). If a¢: lJ(Q) or b ¢: lJ(Q), then C ¢: lJ(Q). Now suppose a, b E lJ(Q). It then follows from I.H. that Sb and Kt"a are calculable and it is decidable whether Ktba<a. By Theorem 25.3 cElJ(Q) holds if, and only if, Ktba<a.

a 4) Suppose C is Qa where a I: O. By I.H. it is decidable whether a E lJ(Q). C E lJ(Q) holds if, and only if, a E lJ(Q).

b 1) Suppose Yl or Y2 is O. If both are the ordinal term 0, then Yl =Y2' Otherwise the ordinal term 0 is less than the other ordinal term.

b 2) Suppose Yi is Yil + ... + Yin, (ni;;:': 1) with principal terms Yil;;:': ... ;;:':Yin, (i=1,2), where n1 +n2>2. Let m :=min(n1,n2). By I.H. Y1k and Y2k are recursively comparable for k=I, ... ,m. If there is a least subscriptj::::;m such that Y1j l:Y2j then Yl <Y2 (Y2 <Yl) if Ylj<Y2j (Y2j<Yl)' Now suppose Y1k=Y2k for all k= 1, ... , m. Then Yl <Y2, Yl =Y2 or Y2 <Yl according as n1 <n2, n1 =n2 or n2 <n1 •

b 3) Suppose Yl and Y2 are principal terms. Then it follows from the I.H. that SYI and SY2 are recursively comparable. If SYI < SY2 (SY2 < SYl)' then Yl <Y2 (Y2 <Yl)· Now suppose SYI =SY2 =v. If both are the ordinal term Qv, then Yl =Y2· If only one of the two ordinal terms is Qv, then this one is less than the other ordinal term. There remains the case Yi=lJocd3i where SYi=SPi=V (i= 1, 2). Then by I.H. oc1 and OC2 are recursively comparable.

238 IX. Higher Ordinals and Systems of lIl-Analysis

b 3.1) OC 1 <OC2 • It follows from the I.H. that K.oc1 is calculable and it is decidable whether K.oc 1 u {Pd <Y2 holds. By Theorem 25.4, Yl <Y2 if K.oc 1 u {Pd <Y2 and Y2<Yl ifY2~K.ocl U {PI}·

b 3.2) oc1 =oc2 · By I.H. PI an<;l P2 are recursively comparable. Yl <Y2, Yl =Y2 or Y2 <Yl holds according as PI <P2' PI =P2 or P2 <Pl·

b 3.3) oc2 <oc1 . Then it follows in the same way as in b 3.1) that Yl and Y2 are recursively comparable.

4. The Subsystem (j( t) of (j( D)

Below we assume O<t E 0*.

Inductive Definition of B(t) c 0*: l. 0 E B(t) 2. O<Y ¢ P, p(y)cB(t) => Y E B(t) 3. oc, P E (j(t), SJ1.(rx)~SP => Brxp E B(t) 4.0<rx<t,rxE{j(t)=>D"EB(t) (j(t) is obviously a subsystem of B(D). Therefore every ordinal OCE{j(t) has a

uniquely determined degree Grx EN.

Theorem 25.6. a) It is decidable,for c E T*, whether c E B(t). b) Any two ordinal terms/rom (j(t) are recursively comparable.

Proof This follows from Theorem 25.5 since (j(t) is a subsystem of (j(D).

Lemma 2. a) rxEB(t)-=P(rx)cB(t) b) rx E (j(t) => Ds" E B(t)

Proof By induction on Goc.

Theorem 25.7. B(t)=qD"0) n O*(D,).

Proof l. Suppose Y E B( t). We prove that Y E q D" 0) and Y < D, by induction onGy.

1.1 Y ¢ P. Then the assertion follows from the I.H. 1.2 Y = Brxp. Then by I.H. rx, P E qD" 0), rx < D, and P < D, hold. It follows

that Y < D, and by Theorem 24.12 b) that Y E qD" 0). l.3 Y = D" where rx # O. Then rx < t and by I.H. rx E qD" 0). It follows that

Y E qD" 0) and y<D,. 2. Suppose Y E q D" 0) and y < D,. Then there exists n such that

yE Cn + 1(D" 0). We prove by induction on n that yE B(t). 2.1 Suppose y E Cn + 1 (D" 0) holds by (D2). Then the assertion follows from

the I.H.

25. A Notation System of Ordinals Based on the O. Functions 239

2.2 Suppose Y E Cn+ 1(Q" 0) holds by (D3). Then y=()a!3 where a<Q" a E C(a, ()ap), p<()ap and a, P E Cn(Q, , 0). By I.H. we obtain a, P E eeL). By Theorems 24.10 and 24.11 there exists {3 such that Sfl(a),:;;S{3=SP and y=ea{3. Here

P= flea) + ri{3) + la{3.

It follows from PE eeL) by Lemma 2 that QspE eeL) and ri{3)E e('c:). If Sfl(a) = S{3, then by Lemma 2 a) we have

If Sfl(a)<S{3 then

In any case we have a, {3E eeL) where Sfl(a)':;;S{3. It follows that y=ea{3E eeL). 2.3 Suppose YECn+ 1(Q,,0) holds by (D4). Then y=Q" where O<a<r. By

I.H. we have a E eeL). Hence it follows that y E eeL).

Theorem 25.8. 8(,) n O*(Q1)=O*(()Q,0). That is to say: The 0 level ofe(,) consists of the segment of all ordinals < ()Q,O.

Proof This follows from Theorems 25.7 and 24.8 b) .

. 5. The Ordinal Ao

By definition e(Q) is the closure of the set {O} under finitely many functions of ordinals. Therefore by Ax. III the set 8(Q) n O*(Q1) has cardinality < Q1' It follows by Ax. IV that there is a least ordinal rt 8(Q) which is < Q1' We denote this least ordinal by Ao. Therefore we have

Lemma 3. IfYEe(Q) there exists ,#0 such that YE8(,).

Proof This follows by induction on Gy.

Theorem 25.9. e(Q) n O*(Q1) = O*(Ao). That is to say: The 0 level of 8(Q) consists of the segment of all ordinals < Ao.

Proof Suppose YEe(Q) n O*(Q1)' By Lemma 3 there exists ,#0 such that y E eeL) n O*(Q1)' By Theorem 25.8 we obtain


Since Ao rf. e(Q), we have OQtO~Ao' Then y E O*(Ao). Thus

From the properties of Ao we also have

6. Relations between Cr (oc) and In (oc)

The set 0 of ordinals considered in §13 can be identified with 0*(Q1)' Then, for the set Cr (oc) of oc-critical numbers defined in §13, we have

/3 E Cr (oc) = oc, /3 E 0*(Q1)'

Lemma 4. OC<OQ10 = oc E C(oc, /3).

Proof By Theorem 24.8 b) from OC<OQ10 we have ocEC(D1,0). Since Q1 is a limit number it follows by Lemma 3 b) of §24 that there is a least ordinalI'{ < Q 1 such that oc E C(I'{, 0). Because of its minimality this ordinalI'{ is not a limit number. If I'{=O, then OCEC(O,O), hence OCEC(OC,f3) by Lemma 3 a) of §24. Otherwise I'{=e+ 1 where oc rf. C(e, 0) and oc E C(e+ 1, 0). Then C(e, 0):;6 C(e+ 1, 0) and consequently by Lemma 9 of §24 e E C(e, 0). Since e<Q1 and OC<Q1' by Theorem 24.8 b) e E C( e, 0) and oc rf. C( e, 0) imply e < OeO ~ oc, and therefore e + 1 ~ oc. Then from oc E C(e + 1, 0) it follows that oc E C(oc, f3) by Lemma 6 a) of §24.

Lemma 5. e<oc~OQ10, y E In (oc) = Oey=y.

Proof There exists /3 such that y = Ooc/3. By Lemma 4 e E C(e, Ooc/3). By Theorem 24.7 b) we obtain Oe(Ooc/3)=Ooc/3, and therefore Oey=y.

Theorem 25.10. Ifoc~OQ10, then: a) Cr(oc)=In(oc)nO*(Q1) b) /3 < Q1 = f/Joc/3 = Ooc/3

Proof of a) by induction on oc. By the definition of Cr (0), Cr (0)=PnO*(Q1)' By Theorem 24.4 a) In (0) = P. Therefore the assertion holds for oc = O. Now suppose 0 < oc ~ OQ1 O. If Y E In (oc) n 0*(Q1)' then for all e < oc, by Lemma 5, we have Oey = y and consequently by I.H. 0ey = y. Then y E Cr (oc) by the definition of Cr (oc). Hence

Now suppose y E Cr (oc). Then for all e < oc, by the definition of Cr (oc) we have f/Jey = y and consequently by I.H. Oey = y. It follows that yEn In (e). If oc is a

~<'"

26. Level-Lowering Functions of the Ordinals 241

limit number, then by Theorem 24.4 a) we have ')!EIn(et). Otherwise et=eto+l where (}eto')! = ')!. By Theorem 24.10 we have ')! ~ In (eto). Since Y E In (eto) it follows that Y E In (et). We therefore also have

(2) Cr(et)cIn(et).

The assertion follows from (1), (2) and Cr (et)c O*(Q1)' b) follows from a), since </Ja and (}a are the ordering functions of Cr (et) and

In (et).

Lemma 6. (}Q 1 0 is the least ordinal Yf such that (}YfO = Yf.

Proof 1. If e<(}Q10 then by Lemma 4 e E C(e, 0). It follows by Theorem 24.8 b) that e < (}eO.

2. If(}Q10~e<Q1' then by Theorem 24.8 a) e ~ C(Q1, (}Q10). By Lemma 6a) of §24 we have

By Theorem 24.7 a) we obtain (}((}Q10)0=(}Q10.

Proof By definition r 0 is the least ordinal Yf such that </JYfO = Yf. By Theorem 25.10 b) and Lemma 6 we have rO=(}Q10.

Theorem 25.12. OT=B(I).

Proof By §13 OT denotes the segment o*(r 0) of the ordinals. By Theorems 25.7 and 25.8

Hence, by Theorem 25.11, OT=B(l).

§26. Level-Lowering Functions of the Ordinals

In this section we introduce some functions of ordinals which we shall use in §27 for the consistency proof of the formal system P A. Thus here we shall be chiefly concerned with functions which assign ordinal terms of lower levels to certain ordinal terms.

242 IX. Higher Ordinals and Systems of l1:-Analysis

l. Basic Concepts

In the following we suppose O<O"~w. We consider ordinal terms from the notation system 0(0"+ 1). The maximal level of these ordinal terms is 0". We also regard the ordinal terms < 0" as natural numbers.

We use the syntactic symbols ct, {3, y, J for principal terms of level 0" in the system 8(0"+ 1), ~,17," /1, v for arbitrary ordinal terms of the system 8(0"+ 1), i,j, k for ordinal terms (natural numbers) < 0". We shall also use these syntactic symbols with subscripts. For brevity we write

(~, 17) for 0~17·

Definition of Si+2~ E 0(0"+ 1): 1. If ~ < Qi+2' then set Si+2~ : =0. 2. If Qi+2~~=~1+"'+~n with principal terms ~l?:"'?:~n (n?:1) then set

Si+2~ : = ~1 + ... + ~m' where m is the largest subscript ~n such that Qi+2 ~ ~m'

Corollaries.!' For each ~ there exists 17<Qi+2 such that ~=si+2~+17. 2. ~<17=Si+2~~si+217. 3. si+2~<Si+217 = ~<Si+217, si+2~+Qi+2~Si+217·

Inductive Definition of h~ : 1. hO=hQa :=0. 2. 0 < i < 0" = hQi : = i. 3. h(~l' ~2):=max{h~1,h~2}' 4. 0< ~ ~ P = h~ : = max {h~i I ~i E P(~)},

that is to say: h~ is the least number such that no Q k occurs in ~ with h~ < k < 0". We say that 17 is a sub term of ~ if either ~ 1=17 E P(~) or ~ =(17, ~2) or ~ = (~1' 17)

or ~=Qq.

Lemma 1. a) h~ < 0". b) S~<O"= S~~h~. c) For every subterm 17 of~, h17 ~h~. d) If 17 is a subterm of ~ such that S17 < 0", then S17 ~h~.

Proofs. These follow at once from the definition of h~.

2. Properties of the Sets of Coefficients

~EM

maximum of the finite set K;*~ u {O}. ~EM


Lemma 2. /fi<k, then: a) K/Kk*¢r:;::,Ki*¢' b) KiKk*¢r:;::,Ki¢' c) Ki*¢=Kk*¢ u Ki*Kk¢'

Proof These follow by induction on G¢.

Lemma 3. a) S1] = (J ~ ¢ < (¢, 1]). b) S¢=(J, h¢<i<(J ~ Ki*¢<¢'

Proof of a). If S1]=(J then by Theorem 25.2 K,,¢«¢, 1]). Here K,,¢=P(¢) since S¢<(J. Therefore P(¢) «¢,1]) holds. Hence we have ¢«¢,1]) since (¢,1]) is a principal term.

Proof of b) by induction on G¢ using Lemma 1 c), d) .. l. ~¢P. For all ¢kEP(¢) such that S¢k<(J we have S¢k<h¢<i and conse

quently K;*¢k= 0. For all ¢kEP(¢) such that S¢k=(J we have h¢k<h¢ and consequently by I.H. Ki*¢k<¢k' Hence we obtain K/¢<¢.

2. ¢=(¢1' ¢2)' Then S¢2 =(J, h¢2 <h¢ and, by I.H., K;*¢2 <¢2 <¢. If S¢1 <(J, then ¢1 < ¢ and S¢1 <h¢ < i, and consequently Ki*¢1 = 0. If S¢1 =(J, then h¢1 <h¢ and consequently by I.H. and a) Ki*¢ 1 < ¢ 1 < ¢. In every case K;*¢ = g d U

K/¢1 UK/¢2<¢' 3. ¢=Q". Then K/¢=K/(J= 0 since S(J=O.

Lemma 4. a) Ki*k'n<k'n. b) K/¢<¢ ~ Ki*Si+2¢<Si+2¢' c) Ki*Si+2k'n<Si+2k'n.

Proofs. a) By Lemma 2 a) K/k'nr:;::,K;*¢. Moreover ki¢ ¢ Ki*k'n. Therefore K/k'n<k'n.

b) If K/¢<Si+2¢' then also K/Si+2¢<Si+2¢' Now suppose Si+2¢<Ki*¢' Then since Ki*¢ < ¢ it follows that there is an ordinal term 1] of least degree such that Si+2¢ < Ki*1] < ¢. By the minimality of G1] we have 1] E P and 1] ¢ K, and therefore 1] = (1] 1,1]2) where ki1] = 1] 1 and

Then Si+2¢=Si+21]1 and consequently Ki*Si+2¢r:;::,K/1]1 <Si+2¢' c) follows from a) and b).

Proof by induction on G1]. If Qi+2 <1], then the assertion is trivial. If 1] ¢ P then it follows from the I.H. If 1] < Qi + l' then K;*1] = 0 < ¢ + 1]. There remains the case 1] = (1]1,1]2) where S1]2 = i+ l. Then by Lemma 2 c)

244 IX. Higher Ordinals and Systems of IIi-Analysis

If ~ E Ki+ 1'11 U{l72}, then G~ <G'1, K{¢cK{'1 <~ +Qi+2 and therefore by I.H. and Theorem 25.2 K;*~ < ~ + ~ < ~ + '1.

By Theorem 25.2 we also have K;*+1'11 <'11' It therefore remains only to prove '11 <~+'1. If '11 <~, then '11 <~+'1. Now suppose ~<I1t. Then from

it follows that there exists '10<Qi+2 such that '11 =~+'10. Hence we obtain Ki + 1 '10 C K i+ 1 '11 < '1, where K i+ 1 '10 = P('1o) since '10 < Qi+ 2' Therefore P('1o) < '1 holds. Since '1 is a principal term, we obtain IJo < IJ. Then IJ 1 = ~ + IJo < ~ + '1 as was to be proved.

Lemma 6. a) ~<Qi+2' Ki*t1'1<'1, ('1, Qi+1)<~ => '1<Ki*~' b) Ki*IJ<IJ, ~«IJ, Qi+1) => K;*~<'1.

Proofs by induction on G~. a) If ~¢P, then the assertion follows from the I.H. Otherwise ~=(~1' ~2) where S~2 =i+ 1. Then by Lemma 2 c)

If IJ<~l' then the assertion IJ<Ki*~ holds. If ~1 <IJ, then from ('1, Qi+1)<~= (~1' ~2)' by Theorem 25.4 we obtain

Then it follows by I.H. that IJ<K;*(Ki+1~lug2}) and therefore also, '1<K;*~. b) If ~ ¢ P the assertion follows from the I.H. If ~ < Qi + l' then K;* ~ = 0 < IJ.

There remains the case ~ = (~1' ~ 2) where S~ 2 = i + 1. Then, by Lemma 2 c)

We therefore only have to prove ~1 <'1. Assume that '1<~1' Since

by Theorem 25.4, we obtain (~1' Qi+l) < K i+ 1 '1. Then there exists '10 E Ki+ 1'1 such that '10 < Qi+ 2 and (~1' Qi+ 1) < IJo· By a)(with '10' ~ 1 instead of~, IJ), using Ki*+ 1 ~ 1 < ~1 we obtain ~1 <K;*IJo~K;*'1<IJ· Hence ~1 <IJ·

26. Level"Lowering Functions of the Ordinals 245

3. The Ordinal Term d/1.

Now, for each number i < (J, we define a map di which assigns a principal term d/1. of level i to each principal term IY. of level (J. We shall use these in §27.

Inductive definition of dilY. : 1. If hlY.~i<(J, set dilY.: =(IY., QJ. 2. If i<hlY., set

d .. ={(Si+2kidi+llY., Q;), if di+llY.=(Si+2kidi+llY., Qi+l) ,IY.. k*d . (Si+2 i i+ llY.+di+llY., Q;) otherwIse.

(The definition is by induction on hlY. - i).

According to this definition dilY. is always a term of the system T*. We prove:

Theorem 26.1. dilY. is a principal term of level i in the system i1( (J + 1).

It suffices to prove that dilY. E i1( (J + 1). If hlY. ~ i < (J, then by Lemma 3 b) K"(IY. < IY.

and consequently dilY.=(IY., Q;) E i1((J+ 1). We prove the assertion by induction on hlY. - i for i < hlY.. In this case dilY. = (~, Q;) where

Then, by I.H., di+ llY. E i1((J+ 1) and consequently ~ E i1((J+ 1) too. By Lemma 4 c)

(1) Ki*Si+2k*di+ llY.<Si+2kidi+ llY.·

Ki*di+ llY. < Si + 2k idi+ llY. + Qi+ 2

also holds. By Lemma 5 we obtain

From (1) and (2) we obtain Ki*~ < ~ and therefore dilY. = (~, Q;) E i1((J+ 1).

Proof 1. If hlY.~i, then dilY. = (IY., Q;) and Qi+l ~IY.. It follows that (Qi + l , Q;)~dilY.· 2. We prove the assertion by induction on hlY.-i for i<hlY.. By the definition

there are the following two cases to be considered. 2.1 dilY.=(~, Q;) where di+llY.=(~, Qi+l)' Then by I.H. (Qi + 2, Qi+1)~di+l!X=

(~,Qi+l)' dilY.=(~,Q;)Ei1((J+I) implies Kn<~· By Lemma 6 from (~,Qi+l)< Qi+2' K;*Qi+2<Qi+2 and (Qi + 2, Qi+l)~(~' Qi+l) we therefore have Qi+2~ K;*(~, Qi+l)' Hence Qi+2~~ and (Qi + 1, Q;)«~, Q;)=dilY.·

2.2 dilY.=(~, Q;) where ~=Si+2kidi+llY.+di+llY.. Then Qi+l ~di+llY.~~. Hence (Qi + l , Qi)~dilY.·

246

Lemma 8. i<j = Kidp.=Kirx u {I}.

Proof 1. If hrx ~j then

Kidp.=Ki(rx, Qj)=Kirx u {I},

since KiQj=KJ= {l}.

IX. Higher Ordinals and Systems of n!-AnaJysis

2. We prove the assertion by induction on hrx - j for j < hrx. 2.1. dp.=(~, Qj) where dj+lrx=(~, Qj+l). Then Kidjrx=Kidj+lrx and conse

quently by I.H. Kidp.=Kirx u {I}. 2.2. dp.=(~, Qj) where ~=sj+2kjdj+lrx+dj+lrx. By Lemma 2 b)

It follows that Ki~=Kidj+lrx and by I.H. Ki~=Kirx u {I}. Then we also have Kidp.=Kirx u {I}.

Proof 1. If hrx ~ i then dirx = (rx, QJ and, by Theorem 25.2, Kirx < dirx. 2. If i < hrx we have dirx = (~, Qi) where, by Theorem 25.2 Ki~ < dirx. 2.1. di+lrx=(~, Qi+l). Then, by Lemma 8,

It follows from Lemma 7 that 1 <dirx. Using Ki~<dirx we obtain Kirx < dirx. 2.2. ~=Si+2kidi+lrx+di+lrx. Then, by Lemma 2 b) and 8)

Therefore from Ki~<dirx we obtain Kirx<dirx.

Lemma 9. If hrx<i+ 1 <(J', then dirx=(Si+2kidi+lrx, Qi) and di+lrx=(Si+2kidi+lrx, Qi+l)·

Proof If hrx<i+ 1 <(J', then dirx = (rx, Qi)' di+1rx=(rx, Qi+l)' Ki*di+1rx={rx} u Ki*rx and by Lemma 3 b) Ki*rx < rx. It follows that kidi+1rx=rx and, since rx is a principal term of level (J' ~ i + 2, we also have Si + 2kidi +1 rx = rx. This yields the assertions.

Proof By the definition of dirx and Lemma 9 dirx=(~, Qi) where

By Lemmata 2 b) and 8


It follows from Lemma 7 that 1 <d;/3. Using the hypothesis KilX< d;/3 we obtain Ki~<diP, We have diP=(Yf, [1i)' We prove ~<Yf. Then usingKi~<diP and [1i < diP, by Theorem 25.4, the assertion dilX < diP follows. By the definition of diP and Lemma 9 it follows that there are two cases for Yf.

1. di+1P=(Yf, [1i+l) and Yf=Si+2Yf. Then Ki*Yf<Yf. Therefore from di+11X< di+1P=(Yf, [1i+l) we obtain by Lemma 6 b) Ki*di+11X<Yf and therefore Si+2kfdi+11X<Yf also. Since di+11X<[1i+2 and Yf=Si+2Yf we obtain -

2. Yf=Si+2kfdi+1P+di+1P. By Lemma 4c)

Ki*Si+ 2kfdi + llX <Si+2k fdi+ l lX· Were di+11X«Si+2kfdi+11X, [1i+l) then by Lemma 6 b) we should obtain Ki*di+1IX<Si+2ktdi+11X which is false. Hence (Si+2ktdi+11X, [1i+1)~di+11X. Using the hypothesis di+1lX<di+1P we obtain (Si+2ktdi+11X, [1i+1)<di+1P. Now di+1P< [1i+2' Therefore by Lemma 6 a) we obtain

Then

too. Using the hypothesis di+11X<di+1P we obtain

Lemma 11. IfIX<P and KilX < diP for all i~j, then dilX < diP holdsfor all i~j.

Proof 1. max U, hlX, hP) ~ i. Then dilX = (IX, [1i) and diP = (P, [1i)' In this case by Theorem 25.4 from the hypotheses IX < P and KilX < diP we obtain dilX < diP by using [1i < diP·

2. j ~ i < max (hlX, hP). Then we obtain dilX < diP by Lemma 10 by induction on max (hlX, hp) - i.

Definition. IX«P (IX is essentially less than P) denotes that IX<P and KilX < diP for alli<u.

Theorem 26.3. If IX« P, then dilX < diP for all i < u.

Proof This holds by Lemma 11 withj=O.

Theorem 26.4. IX« P, P« Y :;. IX« y.

Proof P«Y implies P<y and by Theorem 26.3 diP<diy for all i<u. Using IX«P we obtain IX«Y.

248 IX. Higher Ordinals and Systems of lI:-Analysis

4. The Natural Sum

Definition of the natural sum ~ * rJ : 1. ~*O=O*~ :=~. 2. If ~=(l+"'+(m where (l~"'~(m (m~l) are principal terms and

rJ =(m+ 1 + ... +(m+n where (m+ 1 ~ ... ~ (m+n (n~ 1) are principal terms then set

where n is a permutation of the numbers 1, ... , m+n such that (1t(1)~'" ~(1t(m+n)'

Corollaries. 1. ~ * rJ = rJ * ~. 2. (~*rJ)*(=~*(rJ*(). 3. ~<rJ ~ ~ *( <rJ*'-4. Ki*(~*rJ)=K;*~ u Ki*rJ, Ki(~*rJ)=Ki~ u KirJ·

Theorem 26.5. a) a« (v, a * ~) b) a«f3 ~ (v, a*~)«(v, f3*~). c) fl < v < Q 1, a«(v, (3) => (fl, a) «(v, f3).

Proofs. a) a~a*~«v, a*~) and Kia <;; Ki(v, a*~) hold. By Theorem 26.2 we obtain Kia < di(v, a*~). Hence we have a«(v, a*~).

b) a<f3 implies (v, a*~)«v, f3*~). By a) f3«(v, f3*~). By Theorem 26.4: using a«f3, we obtain a«(v, (3*~) and therefore Kia < di(v, (3*~). By Theorem 26.2 Kiv u Ki~<;;KJv, f3*~) implies Kiv U Ki~<dlv, (3*~). Hence

Hence we obtain (v, a*~)«(v, f3*~). c) Using K(1fl = P(fl) «v, (3) from fl<V<Q 1 and a«v, (3), by Theorem 25.4,

we obtain (fl, a)«v, f3). By Theorem 26.2 from Kiv<;;Ki(v, (3) we obtain Kiv< di(v, (3). Hence we obtain v <di(v, f3), since Kiv=P(v) and di(v, f3) is a principal term. Using fl<V we obtain Kifl = P(fl) <di(v, (3). Using a«(v, f3) we obtain

Hence we have (fl, a)«(v, f3).

5. Deduction Functions

We define some functions which map the set of principal terms of level a into itself. We shall use these in §27.

Inductive definition of the deduction functions of rank p ~ a. 1. The identity function on the set of principal terms of level a is a deduction

function of rank p and degree O.


2. Iff is a deduction function of rank p and degree nand

g(a) = (v,J(a) "" ~),

then 9 is a deduction function of rank p and degree n + 1. 3. Iffis a deduction function of rank p and degree nand

g(a) =(0, Qa+dd(a)) where p~k<u,

then 9 is a deduction function of rank p and degree n + 1. According to this definition every deduction function of rank p#-O is also a

deduction function of every rank < p. Therefore every deduction function is a deduction function of rank O.

If p ~ U we denote the set of numbers i such that p ~ i < u by [p, u). (If p = u this set is empty).

Lemma 12. If 9 is a deduction function of rank p ~ u, a < 13 and Kia < dif3 for all i E [p, u), then g(a) <g(f3) and Kig(a) < dig(f3) for all i E [p, u).

Proof by induction on the degree of the deduction function g. 1. g(a)=a. Then the assertion is trivial. 2. g(a) = (v,J(a) "" ~), where, by I.H., f(a) <f(f3) and K;!(a) < d;!(f3) hold for

a,ll i E [p, u).f(a) <f(f3) implies g(a) <g(f3). By Theorem 26.5 a) f(f3) «g(l3). Hence by Theorem 26.3 we obtain d;!(f3) <dig(f3) and by the I.H. K;!(a) <dig(f3) for all i E [p, u). By Theorem 26.2 from Kiv U Ki~ ~ Kig(f3) we obtain Kiv U Ki~ < dig(f3). Hence. we obtain

for all i E [p, u). I

3. g(a)=(O, Qa+dd(a)) where p~k<u and where, by I.H., f(a) <f(f3) and K;!(a) < d;!(f3) holds for all i E [p, u). By Lemma 11 we obtain d;!(a) < d;!(f3) for all iE [p, u) and therefore, in particular, dd(a)<dd(f3). Hence we obtain g(a) < g(f3). We now prove Kig( a) < dig(f3) for i E [p, u).

3.1. k ~ i < u. Then Kig(f3) = Kiu U {dd(f3)}. By Theorem 26.2 we obtain Kiu U {dd(f3)} < dig(f3). Using dd( a) < dd(f3) we obtain

Kig(a) =Kiu U {dd(a)} <dig(f3).

3.2. p~i<k. Then by Lemma 8



By 3.1. dd(f3) <d,g(f3). Using KJ(f3) < dig(f3) , by Lemma 10 using induction on k - i we obtain dJ(f3) < dig(f3) , Hence

Kig(a) =KiauKJ(a)u{l} <d;9(f3)

Theorem 26.6. a«f3 implies g(a) «g(f3) for every deduction function g.

Proof This holds by Lemma 12 (with p=O).

Lemma 13. If 9 is a deduction function of rank p ~ a, then

for all i<p.

Proof By induction on the degree of the deduction function g. 1. g( a) = a. Then the assertion is trivial. 2. g(a)=(v,f(a):jf~) where, by I.H.,

holds for all i < p. In this case

Hence the assertion follows from the I.H. 3. g(l>:) =(0, Q,,+dd(a)) with p~k<a, where, by I.H.,

for all i<p. Then by Lemma 8

for i < p. Hence the assertion follows from the I.H. We shall use the following theorem in §27.

Theorem 26.7. Ifa=(v, (0, Q,,+dky):jf(j), f3=(v+ 1, (j) andg is a deductionfunction of rank k+ 1 ~a such that y«g(f3), then g(a) «g(f3).

Proof dky<Q,,«v+ 1, (j) implies Q,,+dky«v+ 1, (j). Then

too. Using (j«v+ 1, (j) we obtain


By Theorem 25.2 Kcrv<;;Kcr(v + l)«v+ 1, b). By Theorem 25.4 we obtain

We now prove that g(ex) <g(P) and Kig(ex) <dig(f3) for all ka. 1. Suppose k+ 1 =a. By Lemma 12 ex<fl implies g(ex) <g(fl). 2. Suppose k<i<a. By Theorem 26.2 from Kiv U Kib<;;Kifl we obtain

Ki v U Kib < difl. Using

we obtain

K/f.=Kiv u Kia u {dky} u Kib<difl.

Since this holds for all i E [k + 1, a) using ex < fl we obtain g(ex) <g(fl) and Kig(ex) < dig(fl) by Lemma 12.

3. Suppose i~k. Then Kiex = Kiv u Kia u Kidky u Kib and consequently by Lemma 13

By Lemma 7 a~w«Ql' 0) implies a<dig(f3). Thus

too. From y«g(f3) we obtain Kiy<dig(f3) and by Theorem 26.3 we also obtain dky < dkg(fl). If i = k we obtain

If kk, then by Lemma 8

In every case

By Theorem 26.2

also holds. Kig(ex) <dig(f3) follows from (1)-(4).

252 IX. Higher Ordinals and Systems of llJ-Analysis

§27. The Formal System GPA for a Generalized II~-Analysis

G. Takeuti [4J proved the consistency of subsystems of classical analysis in which the comprehension axiom is essentially restricted to In-formulas. In this section we prove the consistency of a corresponding formal system GPA following the proof of Takeuti.


We use the formal language of classical second order arithmetic described in §20.1 with the syntactic symbols listed there.

Inductive definition of the set Pr (F) of free predicate variables which are' in the scope of a predicate quantifier:

1. If F is a prime formula or a formula U(t), then Pr (F) is empty. 2. Pr (A ~ B): =Pr (A) u Pr (B). 3. If F is a formula Vx%'[xJ or AX%'[XJ(t), then Pr (F) : = Pr (%'[OJ). 4. If F is a formula V X%,[XJ, then Pr(F) is the set offree predicate variables

occurring in F.

Inductive definition of weak formulas. 1. Every prime formula and every formula U(t) is a weak formula. 2. If A and B are weak formulas, then (A ~ B) is a weak formula too. 3. If %'[OJ is a weak formula in which the bound number variable x does not

occur, then Vx%'[xJ and h%'[xJ(t) are weak formulas too. 4. If %'[ UJ is a weak formula in which the bound predicate variable X does

not occur and U rt Pr (%'[UJ), then VX%,[XJ is a weak formula too. G. Takeuti called the weak formulas which we have defined here isolated.

Amongst these formulas there are, in particular, the In-formulas that is to say the formulas VX%,[XJ where no bound predicate variables occur in %'.

By weak predicators we mean the free predicate variables and the predicators of the form AX%'[XJ where %'[OJ is a weak formula.

We call formulas and predicators, which are not weak, strong.

Inductive definition of the degree gr (F) of a formula F: 1. If F is a prime formula or a formula U(t), then set gr (F) : = O. 2. gr (A ~ B) : = max (gr (A), gr (B) + 1. 3. gr (Vx%'[xJ) = gr (AX%'[XJ(t» : = gr (%'[OJ) + 1. 4. IfVX%,[XJ is a weak formula, then set gr(VX%,[XJ) :=gr(%'[UJ)+ 1. 5. IfVX%,[XJ is a strong formula, then set gr(VX%,[XJ) :=gr(%'[UJ)+w. By this definition every formula has a degree w· m + n (where m, n EN). We

call m the weight of the given formula. The weight is 0 if, and only if, the formula is a weak formula.

27. The Formal System GPA for a Generalized nl-Analysis 253

Lemma 1. If P is a weak predicator, then: a) If d[U] is a weak formula, then d[P] is a weakformula too. b) Every formula ff[ P] has the same weight as the formula ff[ U]. c) If VXff[X] is a strong formula, then ff[P] has a smaller degree than

VXff[X].

Proof a) follows by induction on the length of the nominal form d. b) follows from a) by induction on the length of the nominal form ff. c) follows from b) since a strong formula VXff[X] has greater weight than the formula ff[U].

Inductive definition of the rank rk (F) of a formula F: 1. If F is a prime formula or a formula U(t), then set rk (F) : = o. 2. rk (A ~ B) : = max (rk (A), rk (B»). 3. rk (Vxff[x]) =rk (Axff[x](t): =rk (ff[O]). 4. IfVXff[X] is a weak formula, then set rk(VXff[X]) :=rk(ff[U])+ r 5. IfVXff[X] is a strong formula, then set rk(VXff[X]) :=w.

Lemma 2. a) Every formula has a rank ~ w. b) rk (F) < w if, and only if, F is a weak formula. c) The rank of a formula is equal to the maximum of the ranks of its minimal

parts.

1',roof This follows at once from the definition of rank.

Definition of formulas %[ ] and .,q[ , B] : 1. Every N-form % is a nominal form g}[(*1 ~ C)] where g} is a P-form.

Now set %[ ] to be the formula g}[C]. 2. For every NP-form .,q .,q[* l' B] is an N-form %. Now set.,q[ ,B] to be

the formula %[ ].

2. Axioms, Basic Inference and Substitution Inferences

Axioms of the formal system GPA: (Axl) g}[A] if A is a true constant prime formula. (Ax2) %[A] if A is a false constant prime formula. (Ax3) .,q[A, B] if A and B are equivalent simple formulas. (Ax4) ff[a 1, ... , an] (n~ I) if a1, ... , an do not occur in ff and ff[m1' ... , mn] is one of the axioms (Axl}-(Ax3) for all n numerals m1 , ••• , mn •

The minimal parts of the axioms (Axl)-(Ax3) indicated are said to be the principal parts of those axioms.

Remark. The axioms of the system GPA are formulated somewhat more generally than the axioms of the system DA in so far as the principal parts of (Ax3) in DA have degree 0 while those in GPA are only required to be simple. This has no fundamental significance but it proves to be useful for the investigations below.

254 IX. Higher Ordinals and Systems of lIi-Analysis

Basic inferences of the formal system GPA: (Sl) %[iA], %[B] I- %[A ~ B] if B is not the formula_L (S2.0) &'[~[a]] I- &,[\v'x~[x]] if a does not occur in the conclusion. (S2.1) &'[~[U]] 1-&'[\iX~[X]] if U does not occur in the conclusion. (S3.0) ~[t] ~ %[\ix~[x]] I- %[\ix~[x]]. (S3.1) ~[P] ~ %[\iX~[X]]I- %[\iX~[X]] if P is a weak predicator or \iX~[X] is a weak formula. (S4) @''[~[t]] I- C[Ax~[X](t)]. (cut) &,[A], %[A] I- B if &'[%[ ]] f!. B holds. (el) ~[O], ~[a] ~ ~[a'] I- ~[t] if a does not occur in ~. (str) FI- G if Ff!. G holds.

In addition to these basic inferences we use the substitution inferences of the following sort: (sub) ~[U] I- ~[P] if U does not occur in ~ and U ¢ Pr (~[U]).

The way in which the formal system GPA is restricted as opposed to full classical second order arithmetic is because of the restricting condition for (S3.1)inferences. The restriction to weak predicators P corresponds to restricting the comprehension axiom to In-formulas. On the other hand the wider condition on (S3.1) allowing weak formulas \iX~[X] and arbitrary predicators P means the formal system GPA is a strengthening of IIf-analysis. (It contains IIf-comprehension with bar-induction for arithmetic relations.)

We call the basic inferences (Sl)-(S4) principal inferences. The minimal part shown in the conclusion of a principal inference is called the principal part of that inference.

The free variable denoted by a and U in the premises of the inferences (S2.0), (S2.l), (el) and (sub) is called the eigenvariable of that inference.

The formula denoted by ~[P] in the premise of a basic inference (S3.1) is called the eigenformula of that inference. The formula denoted by A in the premises of a basic inference (cut) is called the cut formula.

A basic inference (el) is a complete induction inference and a basic inference (str) is a structural inference.

We set the degree of a basic inference (S3.1) to be the degree of its eigenformula. We set the degree of a basic inference (cut) to be the degree of its cut formula. We set the degree of a basic inference (el) to be the degree of its first premise.

Lemma 3. If the premise of an inference (str) or (sub) is an axiom, then the conclusion of the inference is also an axiom.

Proof The axioms were chosen here to be so general that Lemma 3 would hold.

Lemma 4. If both premises of a cut are axioms which contain no free number variables and the cut-formula is simple, then the conclusion of the cut is also an axiom.

Proof Let &,[A], %[A] I- B where &'[%[ ]] f!. B be a cut satisfying the hypo-

27. The Formal System GPA for a Generalized 111-Analysis 255

theses of the Lemma. It suffices to prove that .?Ji[%[ ]] is an axiom since then by Lemma 3 B is also an axiom. If the minimal part A of .?Ji[A] or of %[A] is not a principal part of the axiom involved then obviously .?Ji[%[ ]] is also an axiom. Otherwise one of the following three cases holds.

1. A is a true constant prime formula and % is an N-form 2[*1' A 2 ] where A and A2 are equivalent. Then A2 is also a true constant prime formula and .?Ji[%[ ]] is the (Axl) .?Ji[2[ ,A2]].

2. A is a false constant prime formula and.?Ji is a P-form 2[A1' *1] where Al and A are equivalent. Then A 1 is also a false constant prime formula and .?Ji[%[ ]] is the (Ax2) 2[A 1 , .¥[ ]].

3 . .?Ji is a P-form 2 1[A 1 , *1] and % an N-form 2 2[ *1' A2] where AI' A and A, A2 are equivalent. Then (by Theorem 20.1) Al and A2 are also equivalent and .?Ji[%[ ]] is the (Ax3) 2 1 [A 1 , 2 2 [ ,A2 ]].

3. Deductions

For the system GPA deductions are defined in the following way as finite tree-like configurations of formulas.

Inductive definition of deductions: 1. If F is an axiom then the formula F forms a deduction with end formula F. 2. If H is a deduction whose end formula is the premise of an (S2.0), (S2.1),

(S3.0), (S3.1), (S4), (str) or (sub) inference with conclusion Fthen the figure

H F

forms a deduction with end formula F. 3. If HI and H2 are deductions whose end formulas are the premises of an

(SI), (cut) or (eI) inference with conclusion F then the figure

forms a deduction with end formula F. We call the formulas of the deduction which do not occur as conclusions of

inferences initial formulas. These initial formulas are axioms. However, besides the initial formulas, a deduction can contain other axioms (as conclusions of inferences).

Inductive definition of the end piece of a deduction: 1. The end formula of the deduction belongs to the end piece of the deduction. 2. If the conclusion of an inference (cut) belongs to the end piece of the

deduction then so too do both premises of this inference.

256 IX. Higher Ordinals and Systems of rrl-Analysis

3. If the conclusion of an inference (str) or (sub) belongs to the end piece of the deduction so too does the premise of this inference.

No other formula belongs to the end piece. By the inferences of the end piece we mean those inferences of the deduction

whose premises belong to the end piece. Every inference of the end piece is a (cut), (str) or (sub) inference.

By the initial formulas of the end piece of a deduction we mean those formulas of the end piece which do not occur as conclusions of (cut), (str) or (sub) inferences. Every initial formula of the end piece is an axiom or the conclusion of a principal inference or a basic inference (el).

Inductive definition of the bundle of formulas of a minimal part M of a formula F of the end piece of a deduction:

1. The minimal part M of the formula F belongs to the bundle of formulas. 2. For every inference (cut) .?JI[A], %[A] f- B such that .?JI[%[ ~] ~ B

occurring in the end piece: If a minimal positive (negative) part of the conclusion B belongs to the bundle of formulas then so too does every identical positive (negative) part of the premise .?JI[A] occurring in .?JI and every identical positive (negative) part of the premise %[A] occurring in %.

3. For every inference (str) A f- B such that A ~ B occurring in the end piece: If a minimal positive (negative) part of the conclusion B belongs to the bundle of formulas so too does every identical positive (negative) part of the premise A.

4. For every inference (sub) .?F[U] f- .?F[P], where U does not occur in .?F, occurring in the end piece: If a minimal positive (negative) partSJ1[P] of the conclusion .?F[P], where U does not occur in $, belongs to the bundle of formulas then so too does each positive (negative) partSJ1[U] of the premise .?F[U].

No other parts of formulas belong to the bundle of formulas. Every bundle of formulas of the kind just defined consists either entirely of

minimal positive parts or entirely of minimal negative parts of formulas of the end piece of a deduction.

Lemma 5. If the minimal part M of a formula of the end piece of a deduction has degree 0 then the bundle of formulas of M consists solely of minimal parts of degree O.

Proof This follows from the definition of bundle of formulas.

Definitions. I. A minimal part of a formula of the end piece of a deduction is said to be explicit if it belongs to the bundle of formulas of a minimal part of the end formula of the deduction. Otherwise it is said to be implicit.

II. The rank of an inference (sub) which occurs in the end piece of a deduction is the maximum of the ranks of the implicit minimal parts of its premise. (If the premise contains no implicit minimal parts then the inference (sub) has rank 0.)

III. A basic deduction is a deduction in which no inference (sub) occurs. IV. A normal deduction is a deduction H for which the following holds:

Every inference (sub) occurring in H is an inference of the end piece of H with a rank < OJ, whose eigenvariable does neither occur in the principal part of an (S2.1 )-inference nor in the eigenformula of an (S3.1 )-inference of H.

27. The Formal System GPA for a Generalized Ill-Analysis 257

V. A formula F is said to be deducible in the formal system GPA if there is a basic deduction with the end formula F.

According to these definitions every basic deduction is a normal deduction and every initial formula of the end piece of a normal deduction is the end formula of a basic deduction.

The main task is now to prove that there is no basic deduction with the end formula L The proof will be obtained in the way of G. Takeuti using reductions of deductions which cannot be restricted just to basic deductions but only to normal deductions.

4. Orders of Normal Deductions

We assign ordinal terms of the notation system fJ(Q) to the formulas of a normal deduction like the ordering functions which W. Pohlers [1] carried out using ordinal terms of the notation system L. For this we first assign weights m < wand ranks (J ~ w to normal deductions in such a way that a normal deduction of weight m and rank (J has the following properties:

1. Every formula of the normal deduction has a weight ~ m and a rank ~ (J. 2. Every inference (sub) which occurs in the normal deduction has a rank

<(J. Obviously for each normal deduction H there is a number m < w such that H

c~m be regarded as a normal deduction of weight m and rank w. If only weak formulas occur in H then H can also be regarded as a normal deduction of a rank <w.

We use principal terms of level (J of the subsystem fJ((J+ 1) of fJ(Q) as orders of the formulas of a normal deduction of rank (J~ w. As in §26 we write (~, 1]) for fJ~1]. For k < (J we also use the function dk introduced in §26 which assigns to every principal term of level (J a principal term of level k. In case (J = 0 set dorx : = rx. If rx, P are principal terms of level (J then, as in §26, rx« P denotes that rx < P and Kirx < diP for all i < (J. The degrees of formulas will be represented by ordinal terms <w2 =(0, (0,0)+(0, 0)). We use v, vo, vl , V2 as syntactical symbols for such degrees.

Inductive definition of the order of a formula in a normal deduction of weight m and rank (J:

1. Every initial formula of the normal deduction has order (0, 0) if (J = 0 and order Q" if (J#O.

2. If the premises of a basic inference (Sl) have orders rx l and rx2 then the conclusion has order (0, rx l # rx 2 ).

3. If the premise of a basic inferenoe (S2.0), (S3.0), (S4) or a basic inference (S2.1) whose principal part is a weak formula has order rx then the conclusion has order (0, rx).

4. If the premise of a basic inference (S2.1) whose principal part is a strong formula has order rx then the conclusion has order (w·(m+ 1), rx) (where m is the weight of the normal deduction).


5. If the premise of a basic inference (S3.1) of degree v has order a then the conclusion has order (v+ 1, a).

6. If the premises of a basic inference (cut) of degree v have orders a l and a2

then the conclusion has order (v, a l =IF ( 2 ).

7. If the premises of a basic inference (CI) of degree v have orders a l and a2 then the conclusion has order (v+ 1, a l =IF(2 ).

8. The conclusion of a basic inference (str) has the same order as its premise. 9. If the premise of an inference (sub) of rank k < (J has order a then the

conclusion has order (0, Q,,+dka). By the order of a normal deduction we mean the order of its end formula.

5. Transformations of Normal Deductions

Theorem 27.1. ifF and G are equivalent formulas then every normal deduction H with end formula F may be transformed into a normal deduction H' of the same weight, rank and order with the end formula G. If H is a basic deduction, so too isH'.

Theorem 27.2. Every normal deduction H with endformula ff[a] where a does not occur in ff may be transformed into a normal deduction H' of the same weight, rank and order with end formula ff[t] for each term t. If H is a basic deduction, then so too is H'.

Both these theorems follow by induction on the length of H.

Theorem 27.3. If H is a basic deduction of weight m, rank OJ and order a with end formula ff[U] where U does not occur in ff then H may be transformed into a basic deduction of the same weight, rank and with an order «(OJ·(m+ 1), a) with end formula ff[P] for each weak predicator P.

Proofby induction on the length of the basic deduction H. We may assume that the eigenvariables of basic inferences (S2.0), (S2.1) and (CI) which occur in Hare chosen so that they are different from U and do not occur in P. From H we then form H' in which the free predicate variable U is replaced by the weak predicator P wherever it occurs in H. It follows from Lemmas 1 and 3 that H' is also a basic deduction of weight m. H' has end formula ff[P]. We still have to prove that H', as a basic deduction of weight m and rank OJ, has an order «(OJ ·(m + 1), a).

1. Suppose ff[U] is an initial formula of H. Then Hand H' have the same order Qro«(OJ·(m+ 1), Qro)'

2. Suppose ff[U] occurs in H as the conclusion of a basic inference (str). Then the assertion follows immediately from the I.H.

3. Suppose ff[ U] occurs in H as the conclusion of a basic inference (S2.1) whose principal part is a strong formula. Then a=(OJ·(m+ 1), ao) where ao is the order of the premise of the (S2.1)-inference. In this case ff[P] is, in H', the conclusion ofa corresponding (S2.1)-inference whose premise has, by I.H., an order po«a. Then H' has order (OJ·(m+ I), Po)«(OJ' (m+ 1), a).

27. The Formal System GPA for a Generalized nl-Analysis 259

4. Suppose ff[ U] occurs as the conclusion of a basic inference other than those under 2. and 3. Then (x=(Vl , (Xl) or (X=(vl , (Xl # (X2)' where Vl <w·(m+ 1), where (Xi is the order of the i-th premise of the basic inference. In this case in H', ff[P] is the conclusion of a corresponding basic inference whose i-th premise has, by LH., an order Pi«(w·(m+l)'(Xi). Then there exists v2 <w·(m+l) such that H' has order

or

6. Reducible Normal Deductions

In specifying parts of a normal deduction we shall write f!. F below to mean that the formula F has order (X in the given place in the normal deduction.

Definition. A normal deduction of order (X is said to be reducible if it can be transformed into a normal deduction whose order is «(X while maintaining its weight, rank and end formula.

Theorem 27.4. If a normal deduction contains a cut with cut formula (A ~ ..L), then the normal deduction is reducible.

Proof Suppose a normal deduction of rank (1 contains the cut

Hl H2 f!. &,[(A ~ ..L)] f1!- %[(A ~ ..L)]

fL B where &'[%[ ]] f!- B

Here Hi and H2 denote the subdeductions of H with end formulas &,[(A ~ ..L)] and %[(A ~ ..L)]. We replace this cut by two cuts with cut formulas ..L and A as follows:

~ %[..L]

H2 f1!- %[(A ~ ..L)] A ~ &'[%[ ]]

~ B where %[(&'[%[ ]] -.. ..L)]f!- B

By this replacement we obtain a normal deduction H' which has the same weight, rank and end formula as H. H' has the axiom %[..1] as an initial formula with order (Xo=(O, 0) (if (1=0) or (Xo = Q.,. (if (1#0). Suppose the formula A has degree v.


Then (A --> 1..) has degree v+ 1 and

There is a deduction function g (according to the definition on p. 249), such that H has order g(y) and H' has order g(b). Since b«y by Theorem 26.6 we have g(b)«g(y). Thus the normal deduction H is reducible.

Theorem 27.5. If a normal deduction contains a basic inference (el) whose conclusion contains no free number variable then the normal deduction is reducible.

Proof Suppose a normal deduction H contains the inference

fL ~[t],

where a does not occur in ~ and t is a numerical term. Here HI and H2 are basic deductions with orders rx and p and end formulas ~[O] and ~[a] --> ~[a']. By Theorem 27.2 for every numeral n, H2 can be transformed into a basic deduction H 2 •n of order p with end formula ~[n] --> ~[n'] which has the same weight and rank as the normal deduction H 2 • Suppose the formula ~[O] has degree v. Then y=(v+ 1, rx#fJ).

We prove then

Lemma. For every numeral n there is a basic deduction H~ of order bn«y with end formula ~[n] which has the same weight and rank as the normal deduction H.

The proof of this Lemma is by induction on n. 1. n=O. Then the assertion holds for HI with bo=rx«y. 2. n =nh. By I.H. there is a basic deduction H~o whose order bno «y. From

this and the basic deduction H2i no' using a cut, we form the basic deduction

f£.n~[n]

Here

Thus the Lemma is proved. The numerical term has a numeral k as its value. Then ~[k] and ~[t] are

equivalent formulas. Therefore by Theorem 27.1 the basic deduction H~ can be transformed into a basic deduction H' of order bk «y with end formula ~[t] where H' has the same weight and rank as the normal deduction H. Now we

27. The Formal System GPA for a Generalized l1;-Analysis 261

replace the subdeduction with end formula ~[t] in H by the basic deduction H' and we obtain a reduction of H.

7. Singular Normal Deductions

Definitions. I. A minimal part ofa formula of the end piece ofa normal deduction is said to be distinguished if the principal part of a principal inference whose conclusion is an initial formula of the end piece belongs to its bundle of formulas.

II. A normal deduction is said to be singular if its end formula is not an axiom and has no distinguished minimal part.

III. A cut .?P[A], JV[A] f- B such that .?P[JV[ ]] ~ B in the end piece of a normal deduction is said to be a suitable cut if the following hold:

1. The negative part A of the premise JV[A] is a distinguished minimal part. 2. If A is a simple formula then the positive part A of the premise '?p[A] is a

distinguished minimal part.

Lemma 6. Every singular normal deduction contains a cut or the conclusion of a basic inference (el) in its end piece.

Proof Suppose H is a singular normal deduction with no cut occurring in its end piece. Then the end piece of H has precisely one initial formula A from which the end formula of H is obtained using only (str) and (sub) inferences. If A were an axiom then by Lemma 3 the end formula of H would also be an axiom. If A were the conclusion of a principal inference then the end formula of H would have a distinguished minimal part. Therefore by the singularity of H it follows that A is the conclusion of a basic inference (el).

Theorem 27.6. Every singular normal deduction whose end piece contains no free number variable is reducible or contains a suitable cut in its end piece.

Proof by induction on the length of the singular normal deduction H. Since the end piece of H contains no free number variable every axiom which occurs in the end piece of H is one of the axioms (Ax1)-(Ax3). If the end piece contains the conclusion of a basic inference (el) then the normal deduction H is reducible by Theorem 27.5. Otherwise by Lemma 6 the end piece of H contains a cut. Then suppose

p:. B where .?P[JV[ ]] ~ B

is the last cut in the end piece of H. Here HI and H2 are normal deductions of


orders IX and P with end formulas 9[A] and %[A] which occur as subdeductions in H. If HI or H2 is singular then the assertion of our theorem follows from the I.H. Now assume that neither HI nor H2 is singular. The end formula of H follows from the formula B by using only (str) and (sub) inferences. Were 9[%[ ]] an axiom then by Lemma 3 the end formula of H would also be an axiom. Therefore by the singularity of H it follows that 9[%[ ]] is not an axiom. Otherwise by the singularity of H it follows that a distinguished minimal part of 9[A] or %[A] can only occur in the cut formula A.

1. Suppose A is a formula'(A l - A2)' 1.1. Suppose A2 is the formula ..l. Then by Theorem 27.4 H is reducible. 1.2. Suppose A2 is not the formula ..l. Then A is a minimal negative part of

%[A] which cannot occur as the principal part of an axiom. Since 9[%[ ]] is not an axiom it follows that %[A] is not an axiom. Since H2 is not singular it follows that the negative part A of %[A] is a distinguished minimal part. Then the cut considered is a suitable cut.

2. Suppose A is a simple formula. 2.1. Suppose neither 9[A] not %[A] is an axiom. Since neither HI nor H2

is singular it follows that A is a distinguished minimal part in both 9[A] and %[A]. Then the cut considered is a suitable cut.

2.2. Suppose 9[A] is an axiom. Since 9[%[ ]] is not an axiom it follows by Lemma 4 that %[A] is not an axiom. Since H2 is not singuhir it follows that the negative part A in %[A] is a distinguished minimal part. It follows by Lemma 5 that A is not a prime formula. Since 9[A] is an axiom and 9[%[ ]] is not an axiom4 is a principal part of the axiom 9[A]. Then, since A is not a prime formula, 9[A] is an (Ax3) ~[Al' A] where Al and A are equivalent formulas and ~[AI' %[ ]] P- B holds. It follows that %[Al] P- B. By Theorem 27.1 the normal deduction H2 can be transformed into a normal deduction with end formula %[Al] while retaining the same weight, rank and order. By introducing an inference (str) we obtain a normal deduction H~ of order p«y with end formula B where H~ has the same weight and rank as H. This shows that H is reducible.

2.3. Suppose %[A] is an axiom. Then as under 2.2 it follows that %[A] is an (Ax3) ~[A, A2] where A and A2 are equivalent formulas and 9[~[ ,A2]] P- B holds. It follows that 9[A 2 ] P- B. Then from HI by Theorem 27.1 using an inference (str) we obtain a normal deduction H~ of order IX«P with end formula B where H~ has the same weight and rank as H. This again shows that H is reducible.

8. Reduction of a Suitable Cut

We use the following notations for parts of a normal deduction H: 1. If F is a formula of H then set H(F) to be the normal deduction with end

formula F which occurs as a subdeduction in H. 2. A deduction thread F - G in the end piece of H is a sequence of formulas

F l , ... , Fn (n~ 1) of the end piece of H where FI is the formula F, Fi occurs in H as the premise of an inference with conclusion Fi + I and Fn is the formula G.

27. The Formal System GPA for a Generalized Ill-Analysis 263

3. A deduction part

of H consists of a deduction thread F-G of the end piece of H and the normal deduction H1 =H(F). Ifsuch a deduction part is replaced by

then this means that Hi is a normal deduction with end formula A -.F and every formula Fi of the deduction thread F-G is replaced by the formula Ai - Fi where Ai is determined as follows: A1 is the formula A. If Fi is a premise of an inference (cut) or (str) then A i + 1 is the formula Ai. If Fi f- Fi+ 1 is an inference (sub) ff[U] fff[P] then Ai+ 1 results from Ai by replacing the free predicate variable U in Ai by the predicator P throughout. If Fn is the last formula G of the deduction thread F-G then B denotes the formula An.

Lemma 7. Suppose F-G is a deduction thread in the end piece of a normal deduction H, H1 =H(F) and H~ is a normal deduction with endformula A - F.lfthe deduction part

H1 Hi F in H is replaced by A - F

I I G B-G,

then the subdeduction H(G) is transformed into a normal deduction with endformula B - G by this replacement.

Proof This is apparent from the form of the inferences (cut), (str) and (sub) which occur as the only inferences in the end piece of H.

We make use of Lemma 7 repeatedly for the following reductions of deductions without particular mention.

Theorem 27.7. If the end piece of a normal deduction contains a suitable cut, then the normal deduction is reducible.

Proof Let H be a normal deduction of weight m and rank (f whose end piece contains a suitable cut.


Case 1. Suppose A ---+ B is the cut formula of a suitable cut. Then H has a deduction part

HI H2

~1 .AI' [-, A ] ~ .AI' o[ Bo] (Sl) ~.AI' or(A o ---+ Bo)]

Ho I fLl &[(A ---+ B)] fL2 .AI'[(A ---+ B)]

(cut) f!. C where &[.AI'[ ]] ~ C

Here the principal part Ao ---+ Bo of the (SI)-inference belongs to the bundle of formulas of the minimal negative part A ---+ B of the second premise .AI'[(A ---+ B)] of the suitable cut. We replace this deduction part as follows:

(str)-------

Ho f1-' &' [A -t BJ f!' ,A -t JV[A -t BJ

(cut)-----------,A-t C

(str) -------

Ho f1-' &' [A -t BJ f!' B -t JV[A -t BJ

Ho (cut)-----------f1-'&'[A-tBJ B-tC

(cut)---------

(cut) ---------------f£.C

There is a deduction function g (see §26, p. 248) with Y2 =g({3) and bi=g(r:t.J (i= I, 2). Since r:t. i«{3 it follows by Theorem 26.6 that bi«Y2 (i= I, 2). Suppose the degrees of the formulas A and B are VI and V2 • Then A ---+ B has degree v: =

max (VI' V2 ) + 1. It follows that

Thus we have that H is reduced by the given replacement.

Case 2. Suppose ,h~[x](t) is the cut formula of a suitable cut. Then H has a deduction part

We replace this deduction part by

27. The Formal System GPA for a Generalized 11~-Analysis

HI f!' 9'O[§"I[t]]

(str)--------f!!-', .j"I[t] -> 9'O[.l.x§"I[X](t)]

(S4)----f!' 9'O[.l.x§"I[X](t)]

265

H2

(S4) f!> % 0[§"2[t]]

(str)--------fh %0[..1.X§"2[X](t)] f!> §"2[t] -> %0[..1.X§"2[X](t)]

I I f!' ,§"[t] -> 9'[..1.x§"[x](t)] tv %[..1.x§"[x](t)]

(cut) --------------IV 9'[.l.x§"[x](t)) fb §"[t] -> %[.l.x§"[x](t))

(cut)-------------,§"[t] -> B §"[t]-> B

(cut) -------------------fiB

There are deduction functions gi such that gi(Pi)=Yi and gi«(X;)=~i· Since (Xi«Pi it follows by Theorem 26.6. that ~i<<Yi (i= 1, 2). Suppose the formula ff[t] has degree v. Then hff[x](t) has degree v+ 1. It follows that

~=(v, (v+ 1, ~l *Y2)*(V+ 1, YI *~2»«(V+ 1, YI *Y2)=Y·

Consequently H is reduced by the given replacement.

Case 3. Suppose Vxff[x] is the cut formula of a suitable cut. Then H has a deduction part .

By Theorem 27.2 the basic deduction HI can be transformed into a basic deduction Hi of the same weight, rank and order with end formula &o[ff I [t]]. We replace the given deduction part as follows:

H' 1

,(str) -------f!' '§"I[t] ..... 9'o['v'x§"IEx]]

H2

(S2, 0) ----fl.' 9'0 ['v'X§"1 [x]]

f!> §"2[t] ..... %0['v'X§"2[X]] (S3, 0) ------- H2

f!> §"2[t] ..... %0['v'X§"2[X]]

f!' ,§"[t] ..... 9'['v'x§"[x]] tv %['v'x§"[x]] IV 9'['v'x§"[x]] fb §"[t] ..... %['v'x§"[x]] (cut) (cut) ------------

,§"[t] ..... B §"[I] ..... B (cut) -----------------


As in case 2 we then have bi«Yi (i= 1, 2) and b«y so that H is reduced by the replacement.

Case 4. Suppose '1Xg;-[X] is the cut formula of a suitable cut. Then H has a deduction part

HI H2

~1 &' o[g;-[ U]] 1 ~2 g;-[P] -4 JV 0['1 Xg;-[X]]

(S2.t) 1" 9'0 [V f ff[ XJ] (S3. ) 1" A' o [T"" [ X]]

( ) fb &,['1Xg;-[X]] fD JV['1Xg;-[X]] cut --------------------~--~~~_=~--

B where &>[ JV[ ]] \-!. B

Here the principal parts of the basic inferences (S2.1) and (S3.1) belong to the bundle of formulas ofthe cut formula '1Xg;-[X]. The formulas in this bundle of formulas are not altered by substitution inferences since the eigenvariable of an inference (sub) does not occur governed by a predicate quantifier. Therefore this bundle of formulas consists solely of identical minimal parts 'v' Xg;-[ X].

Case 4a. Suppose '1Xg;-[X] is a strong formula. Then /31 =(w·(m+ 1), 1X 1) and P is a weak predicator. In this case H has rank w. By Theorem 27.3 the basic deduction HI can be transformed into a basic deduction H~ of weight m, rank wand ordeuo«(w· (m+ 1), 1X 1) = /31 with end formula &' o [g;-[P]]. We replace the given deduction part as follows:

H' 1

(str) ---------------f!-o ,3i'[P] -> &'o[\iX3i'[X]]

H2

(S2.1) --------~1 &'o[\iX3i'[X]]

~ 3i'[P] -> %o[\iX3i'[X]] (S3.l) ------------- H2

~ 3i'[P] -> %o[\iX3i'[X]]

I fb ,3i'[P] -> &'[\iX3i'[XJ] fl.2 %[\iX3i'[X]] fL' &'[V'X3i'[X]] fh 3i'[P] -> %[V'X3i'[X]]

(cut) (cut) -----------------------,3i'[P] -> B 3i'[P] -> B

(cut) --------------------------------------

Here bi«Yi (i= 1, 2). Suppose the formula '1Xg;-[X] has degree v. By Lemma lc) g;-[P] has a degree Vo < v. It follows that

Hence H is reduced by the replacement.

27. The Formal 8ystem GPA for a Generalized nl-Analysis 267

Case 4b. Suppose VXff[X] is a weak formula. It has a rank k+ 1 ~O' where k is the rank of ff[U]. Suppose C is the highest formula below the premises of the suitable cut which has no implicit minimal part of rank > k. Such a formula C exists since the end formula H has no implicit minimal parts. Suppose C has order Yo. We replace the given deduction part as follows:

HI

f!' 9"0[ JOE U]] (str) --------

f!' --,jO[U] --+ 9"o[lIXjO[X]]

--,jO[U] --+ 9"[\lXjO[X]]

H2 ~ jO[P] --+ "¥o[\lXjO[X]]

(83.1) --------

fV "¥[\lXjO[X]] (cut) -----------------

--,jO[U] --+ B

~ --,jO[U] --+ c (sub) H2

--,jO[P] --+ c ~ jO[P] --+ "¥o[\lXjO[X]] (82.1) (cut)

f!' 9"0[\1 XjO[X]]

I --,C--+ "¥[\lXjO[X]] f2' 9"[\1 XjO[X]]

(~D-------------------,C--+ B

fL--,c--+c (str)---

fLc

There is a deduction functionfsuch that f(Pl)=YO and f«(Xl) =150 , Since (Xl «Pi' it follows by Theorem 26.6 that

The newly introduced inference (sub) has rank k<O'. Suppose the formula ff[P] has rank v. Then

No substitution ofa rank ~k occurs between the formulas .Ko[VX[X]] and C.


Therefore there is a deduction function 9 of rank k + I (see §26, p. 248) with g({J2)=YO and g(rt.)=b. By Theorem 26.7 from (1)-(3) we obtain

Thus we have that H is reduced by the given replacement.

9. The Consistency of the System GPA

Definition. A formula F is said to be primitive if every minimal part of F has degree 0 and is therefore a prime formula or a formula U(t).

Theorem 27.8. A primitive formula is deducible in GPA only if it is an axiom.

Proof Suppose H is a normal deduction of order rt. whose end formula is a primitive formula F. We prove by induction on dort. that F is an axiom.

I. Suppose F contains no free number variable. Then we may transform the normal deduction H into a normal deduction H' whose end piece contains no free number variable while retaining its weight, rank, order and end formula. By Lemma 5 the primitive formula Fhas no distinguished minimal part. If Fwere not an axiom then H' would be a singular normal deduction which, by Theorems 27.6 and 27.7, would be reducible. But then it follows from the I.H. that F is an aXiOm.

2. Suppose F is a formula ~[aI' ... , anJ (n?: I) such that no free number variable occurs in~. For each n numerals m I, ... , mn the normal deduction H can, by Theorem 27.2, be transformed into a normal deduction of the same weight, rank and order with end formula ~[mI' ... , mn]. By case I proved above it follows that ~[mI' ... , mnJ is an axiom for all n numerals m I, ... , mn. Therefore F is an axiom (Ax4).

Corollary. The formal system GPA is consistent.

Proof By Theorem 27.8 the formula 1.. is not deducible in GPA since it is a primitive formula and not an axiom.

The consistency proof we have given above depended on transfinite induction up to «ai, Q",), 0) = (}«(}W2(Q", + 1»0.

10. The Subsystem PA of GPA

Let PA be the subsystem of GPA where the inferences (S3.1) and (sub) are only allowed in the following restricted forms: (S3.1) ~[PJ ~ %[VX~[XJJ ~ %[VX~[XJJ, if P is a weak predicator, (sub) ~[UJ ~ ~[PJ, if U does not occur in~,U¢=Pr(~[UJ) and P is a weak predicator.

28. The Semi-Formal System PA* 269

All the Lemmata proved above for GPA also hold for PA. That is to say if (S3.1) is only allowed in the restricted form then the reduction of deductions given in the proof of Theorem 27.7 only requires (sub) inferences in the restricted form.

Substitution inferences can be eliminated in PA since we have:

Theorem 27.9. Every normal deduction in the system PA can be transformed into a basic deduction in the system PA of the same endformula.

Proof Suppose H is a normal deduction in the system PA with end formula F. We regard H as a normal deduction of rank OJ and prove our assertion by induction on the length of H.

1. Suppose F is an initial formula of H. Then the assertion is trivial. 2. Suppose F occurs in H as the conclusion of a basic inference. Then the

assertion follows from the I.H. 3. Suppose F occurs as the conclusion of an inference (sub) g-[U] f- g-[P]

where U does not occur in g- and P is a weak predicator. Then by I.H. we have a basic deduction with end formula g-[U]. From this, by Theorem 27.3, we obtain a basic deduction with end formula F.

Theorem 27.10. The formal system PA is sententially consistent. That is to say: There is no formula A such that A and A ---+ 1.. are deducible in P A.

Proof If A and A ---+ 1.. were deducible in PA then, by Theorem 27.9, there would be basic deductions with end formulas A and A ---+ 1... Thence by a cut we would obtain a basic deduction with end formula 1... But by Theorem 27.8 the formula 1.. is not deducible in PA.

§28. The Semi-Formal System PA*

We now go from the formal system PA to a corresponding semi-formal system PA * in order to eliminate cuts which are not eliminable in PA because of the basic inference (el) which occurs there. PA is embeddable in PA*. In PA* we first eliminate those cuts which have strong cut formulas as the cuts in RA * (§22, p. 203) were eliminated. Then we show the eliminability of the other cuts using the reduction procedure of §27. In this way we obtain a consistency proof for the formal system PA using a weaker transfinite induction than we had to use in §27 for the consistency proof for GPA.

1. Axioms and Basic Inferences of the System P A *

The terms, formulas and predicators of the system PA * are the terms, formulas and predicators of the system PA which contain no free number variable. Hence

270 27. The Formal System GPA for a Generalized l1~-Analysis

every term of the system PA* is numerical and every prime formula is either true orfalse. We define weak and strong formulas and predicators and also the degree and rank of a formula in PA * just as in P A.

Axioms of the system PA *

(Axl) and (Ax2) as in PA. (Ax3*) !LEA, B], if A and B are equivalent formulas of degree O.

Basic inferences of the system PA *

(SI), (S2.1), (S3.0), (S3.1), (S4) and (cut) as in PA.

(S2.0*) &[ff[n]] for every numeral n I- &[\fxff[x]].

All axioms and basic inferences of the system PA * are restricted to formulas in which no free number variables occur. Just as in PA the basic inferences (S3.1) are restricted in PA* to ff[P] -+ %[\fXff[X]] I- %[\fXff[X]] for weak predicators P as opposed to the corresponding basic infere~ces ofGPA.

All the principal parts of axioms of the system PA * are formulas of degree 0 while the principal parts of (Ax3) in PA are only required to be simple. Because of this stronger restriction on the axioms of PA * we gain the permissibility of the structural inferences which we do not take as basic inferences here. Also the basic inferences (CI) occurring in PA do not occur as basic inferences in PA *. Here they arise as permissible inferences.

We count the basic inferences (S2.0*) as principal inferences. The principal part of such an inference is the positive part \fxff[x] indicated in the conclusion.

As before the degree of a cut is defined to be the degree of its cut formula. A cut is said to be weak (or strong) if its cut formula is a weak (or strong) formula.

2. The Strength of a Formula

For eliminating strong cuts we use a notion which we call the strength of a formula. For PA * and also for PA this is defined as follows:

Inductive definition of the strength st (F) of a formula F: 1. If F is a weak formula, then st (F) : = O. 2. If F is a strong formula (A -+ B), then st (F) : = max (st (A), st (B» + 1. 3. If Fis a strong formula \fxff[x] or Axff[x](t), then st (F): =st (ff[O]) + 1. 4. If Fis a strong formula \fXff[X], then st (F): =st (ff[U]) + 1.

According to this definition the strength of a formula'is a natural number which is 0 if, and only if, F is a weak formula.


Lemma 1. If P is a weak predicator then any two formulas $'[U] and $'[P] have the same strength.

Proof This follows by induction on the length of the nominal form $'.

3. Basic Deductions in the System P A *

Below ~, ", , (possibly with subscripts) denote ordinal terms of level 0 in the notation system lJ(w+ 1). By the strength of a cut we mean the strength of its cut formula.

Inductive definition of the basic deductions of strength ~ m. 1. If F is an axiom of the system PA * and ~ is an arbitrary ordinal term of

level 0 of the system lJ(w+ 1), then the formula F forms a basic deduction of strength ~ m and depth ~ ~ with end formula F.

2. If H is a basic deduction of strength ~ m and depth ~ ~ whose end formula is the premise ofa basic inference (S2.1), (S3.0), (S3.1) or (S4) of the system PA* with conclusion F and ~ <" then

H

F

forms a basic deduction of strength ~ m and depth ~" with end formula F. 3. If HI and H2 are basic deductions of strengths ~m and depths ~~i

(i= 1, 2) whose end formulas are the premises ofa basic inference (Sl) or (cut) of strength < m of the system PA * with conclusion F and ~i <" (i = 1, 2) then

forms a basic deduction of strength ~ m and depth ~" with end formula F. 4. If, for every natural number n, Hn is a basic deduction of strength ~ m and

depth ~ ~n with end formula &I[$'[n]] and ~n <" for all n EN then

{Hn}neN &I[V'x$'[x]]

forms a basic deduction of strength ~ m and depth ~" with end formula &I[V'x$' [x]].

According to this definition, in a basic deduction of strength ~ m no cut occurs of strength ~ m. Thus in a basic deduction of strength ~ 1 no strong cuts occur and in a basic deduction of strength ~ 0 no cuts occur at all .

. By the initial formulas of a basic deduction H we mean, as before, those formulas in the deduction H which do not occur as conclusions of basic inferences.


PA * ~ F means: There is a basic deduction of strength ~ m and depth ~ ~ with end formula F.

Then by the definition of basic deduction we have:

PA* Ie, F rrii l '

Theorem 28.1. PA* K¥ .El[A, B] holds for every formula .El[A, B] of the system PA* with equivalent formulas A, B.

Proof It follows by induction on the number of symbols~, 'r:j and A. occurring in A that PA * ~n .El[A, B] (as in the proof of Theorem 22.9). The assertion follows.

Theorem 28.2. IfF and G are equivalent formulas and PA * ~ F holds, then so too does PA* ~ G.

Proof This follows by induction on ~.

Theorem 28.3. The following inverses of (S 1), (S2.0*), (S2.1) and (S4) hold:

(Invl) PA* ~ %[(A ~ B)] ==> PA* ~%[IA], PA* ~ .AI[B]

(lnv2.0*) PA* ~ 8i'['r:jx~[x]] ==> PA * ~ 8i'[~[n]]for every numeral n

(Inv2.l) PA* ~ 8i'['r:jX~[X]] ==> PA* ~ q)'[~[U]] for every free predicate variable U.

(Inv4) PA* ~ C[A.X~[x](t)] ==> PA * ~ C[~[t]].

Proof These follow by induction on ~ since all principal parts of axioms of the system PA* have degree O.

Theorem 28.4. IfPA * ~ F and F ~ G hold for formulas F, G of the system PA * then PA* ~ G also holds.

Proof This follows by induction on ~ using Theorem 28.3 (as in the proof of Theorem 4.4).

4. Embedding of PAin P A *

We define the strength of a cut in the system PA as for PA *. The strength of a basic inference (el) of the system PA is the strength of its first premise ~[O].

Inductive definition of P A Hi; F: 1. If F is an axiom of the system P A, then P A f;}; F. 2. If PA Hi; F holds for the premise of a basic inference (S2.0), (S2.1), (S3.0),


(S3.1), (S4) or (str) of the system PA, then PAJ~+l Fholds for the conclusion F of this basic inference.

3. If PA ~ F (i= 1, 2) holds for the premises F l , F2 of a basic inference (Sl) or (cut) of strength <m or (CI) of strength <m in the system PA, then PA fii F holds for the conclusion F of this inference where n : = max (n l , n2 ) + 1.

Lemma 2. Aformula F is deducible in PA if, and only if, there are natural numbers n > 0 and m such that PA ~ F.

Proof This follows from Theorem 27.9.

Definition. A formula F* of the system PA * is said to be a specialization of a formula F of the system PA if F* arises from F by replacing all the free number variables occurring in F by numerals.

Lemma 3. If F is an axiom of the system PA, then PA * ffJ F* holds for every specialization F* of F.

Proof This follows from Theorem 28.1.

Theorem 28.5 (Embedding Theorem). PA fii F implies PA * f*' n F* for every specialization F* of F.

P~oofby induction on n. 1. Suppose n = 1, then F is an axiom of the system P A. Then the assertion

holds by Lemma 3. 2. Suppose PA fii F by a principal inference, cut or structural inference. Then

the assertion follows by the I.H. and Theorem 28.4. 3. Suppose PA ~ F is obtained by a (CI)-inference. Then F* is a formula

ff*[t*] where t* is a numerical term and we have no<n so .that by I.H. PA* f*.no ff*[O] and PA* f*.no (ff*[k] -. ff*[k']) for every numeral k. Now ff*[k] has strength <m. Using cuts we obtain PA* f*.nO+k ff*[k] for every numeral k. The assertion PA* f*.n ff*[t*] follows by Theorem 28.2 since t* is a numerical term and OJ· no + k < OJ· n.

Corollary to Theorem 28.5. For each formula F deducible in PA there exist ~<OJ2 and mEN such that PA * ~ F* for every specialization F* of F.

Proof This holds by Lemma 2 and Theorem 28.5.

5. Elimination of Strong Cuts in P A *

Lemma 4. IfPA * ~ ff[ U], where U does not occur in ff and P is a weak predicator of the system PA*, then PA* f*H ff[P].


Proofby induction on ,. If ff[U] is an axiom of the system PA*, then ff[P] is a specialization of an axiom of the system P A. Then the assertion holds by Lemma 3. In all other cases it follows by Lemma 1 from the I.H.

Lemma 5. If A is a formula Vxff[x] or VXff[X] of strength m+ 1 where PA* ~+I .?JJ[A], PA* ~+I JV[A] and B is a formula of the system PA* such that .?JJ[JV[ ]] f!. B, then PA * Iii!i+q B.

Proofby induction on '1 (as for Lemma 3 of §22). 1. Suppose JV[A] is an axiom of the system PA*. Then JV[ ] is also an

axiom since A is not a principal part of the axiom JV[AJ. Using .?JJ[JV[ ]] f!. B we have that B is also an axiom of the system PA *. In this case the assertion is trivial.

2. Suppose PA* ~+I JV[A] was obtained by a basic inference (S3.0) or (S3.1) with principal part A from PA* 1;f+1 (Ao ...... JV[AJ) where '10<'1. Here Ao is a formula §[t] or §[P] which has strength m by Lemma 1. By I.H. we obtain

By Theorem 28.3 and Lemma 4 from PA* ~+I .?JJ[A] we obtain

(2) PA* Iii!i .?JJ[AoJ.

The assertion PA* Iii!i+q B follows from (1) and (2) using a cut with cut formula Ao·

3. Suppose PA* ~+I JV[A] by a basic inference other than those under 2. Then the assertion follows from the I.H. (as in the proof of Lemma 3 in §22).

Theorem 28.6. PA* ~+z Fimplies PA* ~~I F.

Proofby induction on ,. 1. Suppose F is an axiom of the system P A *. Then the assertion is trivial. 2. Suppose PA*~+zF was obtained by a.principal inference or a cut of

strength ~ m. Then the assertion follows from the I.H. 3. Suppose PA* ~+z Fwas obtained by a cut with cut formula A of strength

m+ 1 from PA* ~+z .?JJ[A] and PA* I~'+z JV[A] where 'i<' (i= 1,2) and .?JJ[JV[ ]] f!.F, By I.H. we then have PA* ~~I .?JJ[A] and PA* ~~I JV[A].

3.1. Suppose A is a formula (AI ...... A z). Then Al and A z have strengths ~m. By Theorem 28.3 from PA* ~~I JV[(AI ...... A z)] we obtain

(1) PA* ~~\ JV[-,AIJ.

(2) PA* ~~\ JV[A z]'

Suppose '0 : = max ('.J.' 'z)· From PA * ~~\ .?JJ[(AI ...... Az)] and (2) we obtain

(3) PA* ~~:I (At ...... .?JJ[JV[ ]])


by a cut with cut formula A 2 • The assertion PA* Hii~1 Ffollows from (1) and (3) by a cut with cut formula Al since W~2<W~O+ 1 <w~.

3.2. Suppose A is a formula Ax~[X](t). Then it follows by Theorem 28.3 that PA * Hii~lI .9l[~[t]] and PA * Hii~ I .;V[~[t]J. Here ~[t] has strength m. Therefore the assertion PA * Hii~ I F follows by a cut with cut formula ~[tJ.

3.3. Suppose A is a formula \fx~[x] or \fX~[XJ. If ~2 =0, then .;V[A] is an axiom of the system PA * in which A does not occur as principal part. Then .;V[ ] is also an axiom and since .9l[.;V[ ]] f!- F, F is an axiom of the system P A *. In this case the assertion is trivial. Now suppose ~2#0. Then W+W~1+W~2<W~. In this case the assertion PA * Hii~ I F follows from Lemma 5.

Theorem 28.7. ifF is a formula deducible in PA then there exists ~ <eo such that P A * ft F* holds for every specialization F* of F.

Proof By the Corollary to Theorem 28.5 there exist ~o < w2 and mEN such that PA * ~ F* holds for every specialization F* of F. The assertion follows by induction on m.

6. Normal Deductions in the System PA *

By a basic deduction in the wider sense we mean a basic deduction in which arbitrary specializations of the axioms of the system PA may occur as initial formulas besides axioms of the system P A *.

As in PA we employ the following inferences in PA*:

(str) F f- G if F f!- G holds.

(sub) ~[U] f- ~[P] if U does not occur in~, U¢Pr(~[U]) and P is a weak predicator.

We call such an inference an inference of the system PA * if its premise and conclusion contain no free number variables.

Inductive definition of normal deductions: 1. Every basic deduction in the wider sense whose strength is ~ 1 is a normal

deduction. 2. If H is a normal deduction whose end formula is the premise of an inference

(str) or (sub) of the system PA* with conclusion F, then

H

F

forms a normal deduction with end formula F, if in the case of an inference (sub) the eigenvariable does not occur in the principal part of an (S2.l )-inference of H.


3. If H1 and H2 are normal deductions whose end formulas are premises ofa weak cut of the system PA * with end formula F, then

forms a normal deduction with end formula F. As in §27 we define the end piece of a normal deduction, the initial formulas

of the end piece, the bundle of formulas of a minimal part of a formula of the end piece and the explicit and implicit minimal parts of the formulas of the end piece.

In PA *, as in PA, the end piece of a normal deduction consists of only finitely many formulas. But besides its end piece a normal deduction in the system PA * may contain infinitely many formulas.

As before we set the rank of an inference (sub) in a normal deduction to be the maximum of the ranks of the implicit minimal parts of its premise (or 0 if the premise has no implicit minimal part). This rank is already < w here since no strong cut occurs in a normal deduction in the system PA * and therefore every implicit minimal part of a formula of the end piece is a weak formula.

As in PA we assign principal terms of level w from the notation system 8(w+ 1) to the formulas in a normal deduction. We use 0(, f3, y, () (possibly with subscripts) as syntactic symbols for principal terms of level w from the system 8(w+ 1). As before rx«f3 means that 0«f3 and KiO«dif3 for all i<w. Then by Theorem 26.3, diO«di f3 for all i<w. 0(~<f3 means that 0(<<f3 or 0(=f3.

Inductive definition of the order of a formula in a normal deduction: 1. Every initial formula of a normal deduction has order Qro'

2. If the premises of a basic inference (SI) have orders 0(1 and 0(2' then the conclusion has order (0, 0(1 *' 0(2)'

3. If the premise of a basic inference (S2.1), (S3.0), (S4) or a basic inference (S3.1) whose principal part is a strong formula has order 0(, then the conclusion has order (0, O().

4. If the premise of a basic inference (S3.1) whose principal part is a weak formula has order 0(, then the conclusion has order (n + 1,0() where n is the degree of the eigenformula of the inference. (This degree is finite since the principal part of the inference is a weak formula).

5. If every premise ofa basic inference (S2.0*) has an order O(n«O( (where 0( is a principal term oflevel w of the system 8(w+ 1)), then the conclusion has order 0(.

(In this case the order of the conclusion is not uniquely determined by the orders of the premises but can be chosen arbitrarily as long as the condition O(n« 0( is satisfied. Such an 0( always exists as the proof of Theorem 28.8 will show.)

6. If the premises of a weak cut have orders 0(1 and 0(2 then the conclusion has order (n, 0(1 *' 0(2) where n is the degree of the cut formula.

7. If the premise of an inference (str) has order 0(, then the conclusion also has order 0(.

8. If the premise of an inference (sub) of rank k has order rx, then the conclusion has order (0, Qro + dkO().


G H f.!. F (or N H f.!. F) denotes that there is a basic deduction of strength ~ I (or a normal deduction) in which the formula F has order 0(. Since every basic deduction of strength ~ I is a normal deduction if G H f.!. F holds, NH f.!. F also holds.

Theorem 28.S. If PA * ft F holds, then there exists 0( ~< (co, DOl + e) such that GHf.!.F.

Proofby induction on e. I. Suppose F is an axiom of the system PA *. Then the assertion is trivial

since Dw« (co, DOl + e) holds. 2. Suppose PA* ftFwas obtained from PA* f-ii Fi where ei<e (i= I, 2). Then

by I.H. there are O(i~«CO, Dw+ei) such that GHj!i Fi. Using a corresponding basic inference we obtain G H f.!. F where 0( = (n, 0(1) or 0( = (n, 0(1 =11= 0(2) with n < co. Since O(i~< (co, DOl + ei) and ei< e we have O(<«co, DOl + e).

3. Suppose F is a formula &,['v'x~[x]] and PA* W &'[~[n]] where en<e for every numeral n. Then by I.H. there exist O(n~«co, DOl + en) such that GHj!t' &'[~[n]] for every numeral n. Putting 0( : = (co, DOl + e) we have 0(" «0( and GH f.!. F.

7. Reducible Normal Deductions

In the same way as in §27 we define the distinguished minimal parts of formulas in the end piece of a normal deduction, the suitable cuts in the end piece of a normal deduction and singular and reducible normal deductions.

A normal deduction in the system PA * is said to be singular if its end formula is not a specialization of an axiom of the system PA and has no distinguished minimal part.

A normal deduction of order 0( of the system P A * is said to be reducible if it can be transformed into a normal deduction in the system PA* of order «0(

retaining the same end formula.

Theorem 28.9. Every singular normal deduction of the system PA* is reducible.

Proof Suppose H is a singular normal deduction in the system PA *. The proofs of Theorems 27.6 and 27.7 can be immedifltely carried over to H. This leaves only the cases where basic inferences (CI) or strong cuts occur. Indeed in PA* the definition of order for those basic inferences (S2.1) and (S3.1) which have strong formulas as principal parts are somewhat different from those encountered in PA. But this does not affect the possibility of reductions of H since no strong cuts occur in H. H contains no free number variables. It therefore follows from the proofs of Theorems 27.6 and 27.7 that H is reducible.


8. Elimination of Cuts in P A *

Theorem 28.10. NH~ F implies PA* r%0a F.

Proofby induction on dort.. Suppose H is a normal deduction of order rt. with end formula F.

1. Suppose H is singular. Then by Theorem 28.9 there is an rt.o« rt. where NH ~ F. It follows that dort.o < dort. and by I.H. PA * I%oao F and therefore PA* I%0a F.

2. Suppose F is a specialization of an axiom of the system PA. Then the assertion holds by Lemma 3 since w«Qw' O)=doQw~dort..

3. Suppose H is not singular and F is not a specialization of an axiom of the system PA. Then F has a distinguished minimal part.

3.1. Suppose F is a formula %[(A - B)] with distinguished minimal negative part (A - B). Then H has a deduction part

Here the principal part (Ao - Bo) of the (Sl) inference belongs to the bundle of formulas of the minimal negative part (A - B) of the formula %[(A - Bn From H we form two new normal deductions in which the given deduction parts are replaced by deduction parts of the following two sorts:

Then rt.i«rt. (i= 1, 2). It follows by I.H. that PA* l%oa1 (,A - %[(A - B)]) and PA * l%0a2 (B _ %[(A _ Bn Using an (Sl) inference it follows that

PA* Ifr ((A - B)- %[(A - B)]).

The assertion PA* r%0a %[(A _ B)] follows by Theorem 28.4. 3.2. Suppose F is a formula &'[Vx$'[x]] with distinguished positive part

Vx$'[x]. Then H has a deduction part

28. The Semi-Formal System PA*

Hn

(S2.0*) f-ln &> o[$' o[n]] for every numeral n f-l &> o [Vx$' o[x]]

I ~ &>[Vx$'[x]]

279

From H, for every numeral n, we form a new normal deduction in which the given deduction part is replaced by

Then IXn«1X for every numeral n and therefore by I.H. PA* I%0an C-,$'[n]-+ &>[Vx$'[x]]). Using a basic inference (S2.0*) we obtain PA* I%0a (,Vx$'[x]-+ &>[Vx$'[x]]). The assertion PA* ~oa &>[Vx$'[x]] follows by Theorem 28.4.

3.3. Suppose F is a formula %[Vx$'[x]] or %[VX$'[X]] where the indicated part is distinguished. Then H has a deduction part

Ho Ho

f-lo $' oCt] -+ % o [Vx$' o[x]] (S3.0) f-l % o [Vx$' o[x]]

(S3.1) f-lo $'[Po] -+ %o[VX$'[X]] f-l %o[VX$'[X]]

I or I

~ %[Vx$'[x]] ~ %[VX$'[X]]

We form a new normal deduction from H in which the given deduction part is replaced by

Ho Ho

f-lo $' o[t] -+ % o [Vx$' o[x]] f-lo $'[Po] -+ % o[VX$'[X]]

I or I

~ $'[t] -+ %[Vx$'[x]] ~ $'[P] -+ %[VX$'[X]]

Here, in the second case, P as well as Po is a weak predicator. In any case lXo«lX.

By I.H. we obtain

PA* I%0ao ($'[t] -+ %[Vx$'[x]]) or PA* f-!APao ($'[P] -+ %[VX$'[X]]).

The assertion PA * I%0a F follows using a basic inference (S3.0) or (S3.1).

280 IX. Higher Ordinals and Systems of rr:-Analysis

3.4. Suppose Fis a formula &,[V'Xff[XJJ, &'[hff[xJ(t)J or %[hff[xJ(t)] where the minimal part indicated is distinguished. Then we have in the same way as before (Xo«(X so by I.H. we have:

PA* ~<zo (-...,ff[UJ ---+ &,[V'Xff[XJJ, PA* ~<zo (-,ff[tJ ---+ &'[hff[xJ(t)]) or PA * fW'<zo (ff[tJ ---+ %[hff[xJ(t)], where, in the first case, U does not occur in ff.

Using a basic inference (S2.1) or (S4) and Theorem 28.4 we obtain the assertion PA* ~o<Z F.

Theorem 28.11. ifF is a formula deducible in PA, then there exists 11 < «cu, Q",+ eo), 0) such that PA * 1-6- F* holds for every specialization F* ofF.

Proof By Theorem 28.7 there exists e < eo such that PA * If F* holds for every specialization F* of F. Then by Theorem 28.8 there exists (X~«cu, Q",+~) such that NH'r- F*. Here (X may depend on the choice of the specialization F*. It follows from Theorem 28.10 that PA* ~o<z F*. Then if 11 : = do(cu, Q", + e) we have

and therefore, also, PA * 1-6- F* where 11 depends only on F.

Corollary. The formula ..1 is not deducible in P A.

Proof This follows from Theorem 28.11 since it follows by induction up to «cu, Q",+eo), 0) that there is no 11 < «cu, Q",+eo), 0) such that PA* 1-6-..1.

At the same time we have also proved the sentential consistency of the system PA since

A, -,A 1-..1

is a permissible inference of the system P A. This consistency proof uses transfinite induction up to «cu, Q", + eo), 0). This

is weaker than the transfinite induction wt( used for the consistency proof of PA in §27 since «cu, Q", +eo), 0) < «cu2 , Q",), 0).

Remark. The normal deductions of the system PA * we have defined here correspond to normal deductions of rank cu in the system PA. In a similar way to that for PA we may define normal deductions of rank < cu in PA * and the corresponding Theorems hold for them.

From Theorems 28.6, 28.8 and 28.10 we have that ifPA* ~Fthen there is an ordinal term 11 such that PA * 1-6- F. If we restrict F to deducible formulas in the system PA then, by Theorem 28.5, e may be restricted to ordinal terms <cu2 while we can only restrict the corresponding ordinal term 11 so that 11 < «cu, Q w + eo), 0). But it is possible to estimate a certain ordinal term bound , such that to every

29. Proofs of Well-Ordering 281

ordinal term ~ <, such that PA * ~ F there is an ordinal term '1 <, such that PA*ij- F. The smallest bound of this kind which we obtain by our theorems is «ro,Q",+Q1),0)=suP~n where ~o:=O and ~n+1:=«ro,Q"'+~n)'0). Using the

neN

next theorem we prove that the ordinal term «ro, Q", + Q1)' 0) is an ordinal term bound of the type described.

Theorem 28.12. The following three properties of a formula F are equivalent to each other.

(1) There exists meN and ~«(ro, Q", +Q1)' 0) such that PA* ~ F. (2) There exists a«(ro, Q",+Q1) such that NH~F. (3) There exists '1 < «ro, Q", +Q1)' 0) such that PA* ~ F.

Proof of (1) => (2). Obviously the ordinal term «ro, Q",+Q1)' 0) denotes an 6-

number. Therefore it follows from (1) by Theorem 28.6 using induction on m that there exists '1< «ro, Q",+Q1)' 0) such that PA* ptF. Then by Theorem 28.8 there exists a,«ro,Q",+'1) such that NH~F. It follows from '1«(ro,Q",+Q1)'0) that (ro, Q",+'1)«(ro, Q",+Q1) and therefore a«(ro, Q",+Q1) too.

Proof of (2) => (3). By Theorem 28.10 it follows from (2) that PA * Iff« F. Then it follows from a«(ro, Q",+Q1) that doa«(ro, Q", +Q1)' 0).

This completes the proof of Theorem 28.12 since (3) => (1) holds trivially.

This theorem says that: If one defines a formula to be deducible in PA * if, and only if, there exist meN and ~«(ro,Q",+Q1)'0) such that PA*~Fthen every formula F deducible in PA * is deducible with a cut free proof (that is to say, there exists '1 < «ro, Q",+Q1)' 0) such that PA* ~ F) and Fis deducible in PA* if, and only if, there exists a«(ro, Q", + Q1) such that NH ~ F. Moreover, every specialization of a formula deducible in PA is also deducible in PA *.

§29. Proofs of Well-Ordering

1. A Constructive Proof of Well-Ordering for Subsystems of 8(Q)

In the following we let -r be an ordinal term > 0 of level 0 in the notation system 8(Q).

Assumption. The well-ordering of the ordinal terms <-r is constructively provable.

Under this assumption we produce a constructive proof of well-ordering for the subsystem 8(-r) of 8(Q). In particular this assumption is valid for -r~ro+ 1 so we prove constructively that the systems of ordinal terms used in §27 and §28 are well-ordered.

282 IX. Higher Ordinals and Systems of JIl-Analysis

However the well-ordering of the full system lJ(Q) is not deducible using the weak methods of proof which we employ in this section. W. Buchholz [2] has produced a proof of the well-ordering of lJ(Q) and certain stronger systems of ordinal terms using stronger methods of proof in a certain constructive way. The proof of well-order given below uses special techniques from W. Buchholz' proof.

Notations for M~lJ(Q): 1. Let M(rx) be the set of ~ E M which are <rx. 2. Let W(M) be the set of rx E M such that M(rx) is well-ordered.

Lemma 1. If M, M* are subsets oflJ(Q) then: a) rx E W(M) => M(rx) ~ W(M). b) rxEM, M(rx)~ W(M) => rxE W(M). c) rx E M(f3) (\ W(M*), M(f3) ~ M* => rx E W(M).

Proof This follows from the definition of W(M).

Inductive definition of Ma for 11<-': Let Mn be the set of rx E lJ(Q) such that Srx~11 and K"rx£;; W(Mv) for all V<I1.

Lemma 2. If 11<-', then: a) rx E Ma => P(rx)£;;Ma b) rx,f3EMa=>rx+f3EMa c) rx+f3EMa=>f3EMa d) rx, f3 E W(Ma) => rx+ f3 E W(Ma) e) rx E W(Ma) ¢> P(rx) ~ W(Ma)

Proofs. a) follows immediately from the definition of Ma· b) rx, f3 E Ma implies S(rx + (3) = max (Srx, S(3) ~ 11 and Kv(rx+ (3) ~ Kvrx uKvf3~

W(Mv) for all V<I1. Hence rx+ f3 E M a. c) By a) rx+f3EMa implies P(f3)~P(rx+f3)~Ma. Obviously OEMa. By b),

using P(f3)£;;Ma we obtain f3 E M a· d) Because of the hypothesis f3 E W(Ma) we can use induction on Ma up to

f3. Then we have the I.H.

Now suppose ~ E Ma(rx + (3). IH E Mirx), then ~ E W(Ma) follows from rx E W(Ma) by Lemma 1 a). Otherwise ~=rx+'1EMa where '1<f3. Then we obtain '1EMa(f3) by c) and ~ = rx + '1 E W(Ma) by the I.H. Hence Mirx + (3) £;; W(Ma)· rx, f3 E W(Ma) imply rx, f3 E Ma and by b) rx + f3 E Ma. By Lemma 1 b) we obtain rx + f3 E W(Ma).

e) rx E W(Ma) implies rx E Ma and by a) P(rx) £;; Ma. By Lemma 1 a) we obtain P(rx) ~ W(Ma). Obviously 0 E W(Ma). Therefore from P(rx) ~ W(Ma) we obtain rx E W(Ma) by d).


Lemma 3. If p:::;O"<-c: a) M,iQp+ 1) r;;;.Mp. b) W(Mp)r;;;. W(Mo.).

Proof a) follows immediately from the definition of Ma. b) is obtained as follows by induction on 0". Suppose (X E W(Mp). Then (X E Mp

and S(X:::;p. If v<p then Kv(Xr;;;. W(M). If p:::;v<O" then Kv(X=P«(X) and by I.H. (xE W(Mv) and therefore by Lemma 2 e) Kv(Xr;;;. W(Mv). Hence we have (X E M a(Qp+1). Using (X E W(Mp) it follows by a) and Lemma 1 c) that (X E W(Ma). Hence W(Mp)r;;;. W(Ma).

Definition. Qt: = U W(Ma)· a<t

Lemma 4. IfO"<-c then Q.(Qa+1)= W(Ma)·

Proof Suppose (X E Q.(Qa+ 1). Then S(X:::; 0" and there exists v < -c such that (XE W(Mv). It follows that (xE Mv and S(X:::;v. If S(X<v then P«(X) = KSa(Xr;;;. W(MSa) and by Lemma 2 e) (X E W(Msa). Hence in every case (X E W(Msa). Using S(X:::; 0" it follows by Lemma 3 b) that (XE W(Ma). Therefore Q.(Qa+1)r;;;. W(Ma) holds. By definition we also have W(Ma)r;;;.Q.(Qa+1).

Theorem 29.1. Qt is well-ordered.

Proof IhE Qt then S(X<-c and by Lemma 4 (xE W(MSa) = Q.(QSa + 1)· By Lemma 1 c) we have (X E W(Qt). Hence Qtr;;;. W(Qt) and therefore Qtis well-ordered.

Theorem 29.2. P«(X)r;;;.Qt = (xE Qt.

Proof P(ct)r;;;.Qt implies S(X<-c and by Lemma 4 P«(X)r;;;. W(Msa). It follows by Lemma 2 e) that (XE W(Msa)r;;;.Qt.

Theorem 29.3. IfO<O"<-c then QaE Qt.

Proof By the assumption on -c, 0" E W(Ma). It follows from Lemmata 2 e) and 3 b) that for all v < 0"

Hence Qa E Ma. S(x<O" for all (X E MiQa) and P«(X) = KSa(X r;;;. W(MSa consequently by Lemmata 2 e) and 3 b) (X E W(Ma). Hence Ma(Qa) r;;;. W(Ma). Using Qa E Ma it follows by Lemma 1 b) that QaE W(Ma)r;;;.Qt.

Lemma 5. If 9(Xp E 9(Q) where Sp < -c, then:


Proof If v < S/3 then K.(KsplX) = K.IX and consequently K}JIX/3 = K.IX U K./3 = K.(KsplX U {/3}). Therefore it follows from lJlX/3 E Msp that K.(KsplX U {/3}) £: W(M.) for all v < S/3. Hence we obtain KsplX U {/3} £: Msp.

Lemma 6. IX E Q, implies K.IX £: Q, for all v < T.

Proof IX E Q, implies SIX<T and by Lemma 4 we obtain IX E W(MSa) and therefore IX E MSa. It follows that K.IX £: W(M.) £: Q, for all v < SIX. If SIX ~ v < T then K.IX =

P(IX). Then it follows from IXE W(MSa) by Lemma 2 e) that K.IX£: W(MSa)£:Qt.

Lemma 7. IX, /3 E Q" lJlX/3 E lJ(Q), MsP(lJlX/3) £: Qt => lJlX/3 E Q,.

Proof IX, /3 E Qt implies S/3 < T and by Lemmas 6 and 4 we obtain

for all v < S/3. Hence we have lJlX/3 E Msp. By Lemma 4 MsP(lJlX/3) £: Qt implies MSP(lJlX/3) £: W(Msp)· Using lJlX/3 E Msp we obtain lJlX/3 E W(Msp) £: Q, by Lemma 1 b).

Theorem 29.4. IX, /3 E Q" lJlX/3 E lJ(Q) => lJlX/3 E Q,.

Proof We have

(1) IX, /3 E Q, (2) lJlX/3 E lJ( Q)

as hypotheses and lJlX/3 E Q, as assertion. By Theorem 29.1 it follows from (1) that we can use induction over Qt on IX. Then we have the induction hypothesis

We may also use a subsidiary induction over Qt on /3 with induction hypothesis

'1 E Q, , '1 < /3, lJlX'1 E lJ( Q) => lJlX'1 E Q, .

If S'1 = S/3 it follows from (2) by Theorem 25.3 that lJlX'1 E lJ(Q) too. Therefore we have, in particular, as induction hypothesis

In order to prove the assertion lJlX/3 E Qt it suffices, by (1), (2) and Lemma 7, to prove that MsP(lJlX/3)£: Qt. Now suppose

(5) Y E MsP(lJlX/3).

Now in order to prove our theorem it suffices to prove Y E Qt under the hypotheses (l)--{5).

29. Proofs ofWeJl-Ordering 285

Proof of Y E Qt by induction on the degree Gy. 1. Y ~ P. If () E P(Y) then G{) < Gy and by (5) and Lemma 2 a) () E Msp(BrxP). It

follows by I.H. that () E Qt for all () E P(Y). By Theorem 29.2 we have Y E Qt. 2. Y = Q". where 0 < CT. Since Y < Brxp and P E Qt' CT ~ Sp <. and therefore

Y E Qt by Theorem 29.3. 3. y=BY1Y2 where y~Ksprx u {P}. It follows from (1) by Lemma 6 that

Ksprx u {P} £ Qt and by Lemma 4 that Ksprx u {P} £ W(Msp). Therefore there exists e E W(Msp) such that Y ~ e. Then by (5) Y E Msp. By Lemma 1 a) we obtain yE W(Msp) £ Qt·

4. y=BY1Y2 where Ksprx u {P}<y. Then Sp~Sy=S'Y2- Since by (5) y<BrxP, we have SP=Sy. It follows by Theorem 25.2 that KSPY1 u {Y2} <yo By I.H. we have () E Qt for all () E KSPY1 u {Y2} and therefore

(7) Y2 E Qt

4.1. Y1 =rx. Now y=BY1Y2 <BrxP and SP=SY2 imply QSP~Y2 <po Using (7) we therefore have, by (4), y={JrxY2 E Qt.

4.2. Y1 =lrx. By Theorem 25.4 Ksprx u {P}<y=BY1Y2<(JrxP implies Y1 <rx. We prove K".y 1 £ Qt for all CT <. by induction on CT.

a) CT~ Sp. Then K"'Y1 = K".(KsPY1). Using (6) we obtain K"'Y1 £ Qt by Lemma 6. b) SP<CT< •. By I.H. for all V<CT, we have Kv(K".Y1)=KvY1£Qt and conse

quently by Lemma 4 Kv(K"'Y1)£W(Mv). Hence we obtain K"'Y1£M".. By §24 (p. 232) Brxp E B(Q) implies SJl(rx) ~ Sp < CT and Orx(Q". + 1) = BrxQ". E B(Q). Similarly BY1Y2 E B(Q) and SP=SY2 imply OY1(Q".+ 1)=BY1Q".E B(Q). By Theorem 25.2 we obtain K"'Y1 <By1Q".· By Theorem 24.4 b) Y1 <rx implies OY1(Q".+ 1)~Orx(Q".+ 1) and therefore BY1Q".~BrxQ".. Hence we have K"'Y1 £M".({JrxQ".). Q". E Qt by Theorem 29.3. Thus the hypotheses (1)--(5) are satisfied with rx, Q"., () E K"'Y1 replacing rx, p, y. If () E K"'Y1 then G{) ~GY1 < Gy. Therefore by induction on Gy it follows that () E Qt for all () E K".y 1 and therefore K".y 1 £ Qt·

Since K"'Y1 £ Qt for all CT<. and since Y1 < rx E Qt too we have in particular P(Y1)=Ksy ,Y1£Qt. By Theorem 29.2 we have Y1EQt. Using (7), Y1<rx and BY1 Y2 E (J(Q) by (3) we obtain Y = (JY1 Y2 E Qt·

Theorem 29.5. (J(.) is well-ordered.

Proof Ifrx E B(.) we obtain rx E Qt by induction on Grx because of Theorems 29.2-29.4. Hence B(.) £ Qt. By Theorem 29.1 it follows that B(.) is well-ordered.

2. The Formal System IOn of n-Fold Iterated Inductive Definitions

The proof of well-ordering of B(.) above is not predicative nor can it be presented predicatively since (J(I) already contains all ordinal terms < r o. However the proof does not require full classical second order arithmetic but can be


accomplished in a formal system which comes from first order arithmetic by adding generalized inductive definitions. In this way the proof of well-ordering does have a constructive character.

The proof of well-ordering of ()(N) can be formalized in a formal system IDN for 0 < N < 0) where ID" is constructed as follows for 0 ~ n < 0).

The terms and prime formulas of the system ID" are the same as those of the formal systems GPA and P A.

Inductive definition of the formulas and predicators of the system ID,,: I. Every prime formula is a formula. 2. Every free predicate variable is a predicator. 3. If P is a predicator and t a term, then pet) is a formula. 4. If A and B are formulas so too is (A ~ B). S. If ff[O] is a formula in which the bound number variable x does not occur

then Vxff[x] is a formula and Axff[x] a predicator. . 6. Ifn>O and

Vx(U(x)~ V(x»~ Vx(d[U, x] ~ d[V, x]) (Monotonicity condition for d)

where U, V are distinct free predicate variables and no free number variable and no free predicate variable occurs in d, is a formula deducible in ID"_l then f1Jd is a predicator.

No bound predicate variables occur in IDn. We use the same syntactic symbols for IDn as for GPA.

Axioms of the system IDn: (Axl)-(Ax4) as for GPA (AxS) d[PI'd' t] ~ PI' .• ,(t) (Ax6) Vx(d[P, x] ~ P(x» ~ (PI' d{t) ~ pet»~

Basic inferences in the system IDn: (Sl), (S2.0), (S3.0), (S4), (cut), (el) and (str) as for GPA, (sub) ff[U] I- ff[P] if U does not occur in ff.

All the axioms and basic inferences of the system ID" shall consist of the formulas of ID" defined above. P denotes an arbitrary predicator of the system IDn and Pl'd a predicator given by rule 6 of the definition above. By (AxS) and (Ax6) such a predicator must represent the predicate f1J d such that Pl'd(n) holds if, and only if, n is an element of the smallest set {x I P(x)} such that Vx(d[P, x] ~ P(x» where the existence of such a smallest set is guaranteed by the monotonicity condition imposed on d. In this way a generalized inductive defifiition is formalized by Pl'd.

ID" I- F denotes that F is a deducible formula in ID".

Theorem 29.6. IDn I- F=> IDn+l I- F.

Proof This follows immediately from the definition of IDn.


3. Formalization of the Proof of Well-Ordering of B(N) in IDN

Let n be the bijective map of N x N onto N defined on p. 96. As before we set Po:=2 andpn' for n#O, to be the n-th odd prime.

Inductive definition of an injective map rx f--?-' rx' of B(Q) into N: l. '0': = 1 2. IfO<rxEB(Q) set 'Qa' :=P,,(,a,.o) 3. If Brxf3 E B( Q) set' Brxf3' : = P ,,(,a,., P,)

4. If rxl+···+rxnEB(Q) where rxl~···~rxn (n>I) are principal terms set n

, rx 1 + ... + rxn' : = TI 'rx i '.

i= 1

Corollary. If rx E B( Q) then 0 <, rx' EN. Hence it follows that rx f--?- ' rx' is injective.

Definition of a surjective mapping /1 f--?- n of N onto B(Q)u {Ao}. (Ao denotes the least ordinal which is not denoted by an ordinal term in the system 8(Q).)

n:={rxE.B(Q) if. 'rx'=n, Ao otherwise.

Obviously n is calculable for every natural number n. Ifm, /1 EN then -< (m, /1) holds if, and only if, m< n. Therefore -< is a decidable

2-place arithmetic predicate. As before we write s -< t for -< (s, t). If:F is a I-place nominal form such that :F[t] is a formula for every term t

then the progressiveness of a predicator P with respect to the relation -< on the set {x I :F[x]} is expressed by the formula

Prog [:F, P] : = V'x(:F[x] /\ V' y(y -<x /\ :F[y] --> P(y» --> P(x».

If

IDn f-:F[t] /\ Pr [:F, P] --> pet)

holds for every predicator P then this means that the well-ordering of the set {x I :F[x]} by -< up to t is provable in IDn'

From now on we let N denote a fixed natural number> O. We define decidable arithmetic predicates as follows:

1. TN(m) holds if, and only if, mE B(N). 2. Si(m) holds if, and only if, mE B(Q) and Sm~ i. 3. K;(m, /1) holds if, and only if, n E B(Q) and mE Kin.

n

Define (A /\ B) and (A v B) as above (p. 25). We write A 1\ Bi for A /\ Bo i=O

n

/\ ... /\ Bn and V AJor Ao v ... v An' i=O

288 IX. Higher Ordinals and Systems of rr:-Analysis

Inductive definition of formulas Jtn[t], dn[U, t] and 1I'"n[t]: 1. Jt o[t] : = So(t) 2. dn[U, t] : = Jtn[t] 1\ 'v'y(Jtn[y] I\y-<t~ U(y» 3. 1I'"n[t] : = Jtn[t] M!I sln(t)

i=O

By induction on n one simultaneously proves: a) Jt nEt] is a formula of the system IDn. b) IDn f- ('v'x(U(x) ~ Vex»~ ~ 'v'x(dn[U, x] ~ dn[V, x]). c) f!J sin is a predicator of the system IDn + 1.

d) 1I'"n[t] is a formula of the system IDn + 1.

It follows that Jtn[t], f!J sln(t) and 1I'"n[t] are formulas ofIDN if n <N. Therefore, as (Ax5) and (Ax6) hold for n<N

IDN f- Jtn[a] 1\ 'v'y(Jtn[y] 1\ Y -<a ~ f!J sln(Y» ~ f!J slJa) IDN f- 'v'x(Jtn[x] 1\ 'v'y(Jtn[y] I\y-<x~ P(y»~ P(x»~ (f!J sln(t)~ pet»~

It follows by the definition of 1I'"n[t] that

IDN f- Prog [Jtn, Ax1l'"n[X]] IDN f- 1I'"n[t] ~ Jtn[t] 1\ (Prog [Jtn, P] ~ pet»~

From here and the definition of Jto[t] and Jtn+1[t] we therefore see that the sets Mn and W(Mn) defined on p. 282 are expressed in IDN for n < N by

Now set

N-l

,qN[t] : = V 1I'";[IJ. i=O

Then ,q N[t] is a formula of the system IDN such that the set QN defined on p. 283 is represented by

The proof of well-ordering of (}(N) can now be translated into IDN in the following way:

Theorem 29.7. IDN f- Prog [TN' P] ~ 'v'x(TN(x) ~ P(x»

Proof As for Theorem 29.1 one proves


In the same way as in the proof of Theorem 29.5 we proved B(N) ~ QN we now prove

From (2) we have

IDN I- Prog [TN' P] ~ Prog [!2 N' h(TN(x) ~ P(x))]

Since (1) also holds with h(TN(x) ~ P(x)) instead of P we obtain

Using (2) we obtain

IDN I- Prog [TN' P] ~ (TN(a) ~ Pea)).

The assertion follows using a basic inference (S2.0).

4. Embedding IDn in a Subsystem of PA

If 0 < v ~ ill let PAv be the subsystem of PA which contains only formulas of rank < v and let PA? be the subsystem of PAv in which no free predicate variable occurs inside a predicate quantifier. That is to say: The set Pr (F) defined on p. 252 is to be empty for every formula Fin the system PA? Then PAw is the subsystem of PA which contains only weak formulas.

If 0 < n < v ~ ill then PAn is a subsystem of PAv and PA~ a subsystem of PA?

Lemma 8. If g-[U] is a formula and P a predicator in the system PA? (O<V~ill) where U does not occur in :!F, then

:!F[ U] I- :!F[P]

is an inference (sub) in the system PA?

Proof In this case Pr (:!F[U]) is empty and P is a weak predicator. Therefore the rank of :!F[P] is equal to the maximum of the ranks of :!F[U] and P(O). The assertion follows.

Inductive definition of a formula P and a predicator P in the system PA~ + 1 for each formula F and predicator P of the system IDn.

1. For each prime formula F set P : = F. 2. For each free predicate variable U set U: = U. 3. For each predicator P in the system IDn set P(t): =P(t). 4. (A~B):=(A~B).

290 IX. Higher Ordinals and Systems of nt-Analysis

s. If ff[a] is the formula ff[a] where a does not occur in the nominal forms ff and ff then set

6. If f?JJ..t is a predicator in the system IDn and d[U, a] is the formula d[U, a] where U and a do not occur in the nominal forms d and d then n > 0 and, because of the inductive definition, dEl!, a] is a formula of the system PA~. Then set

f?JJ..t : = AzVY(Vx(d[Y, x] -+ Y(x» -+ Y(z».

Theorem 29.8 (Embedding Theorem). If IOn I- F holds then F is a deducible formula in the system PA~ + 1.

Proofby induction on n with a nested induction on the length of the deduction of Fin IOn.

1. Suppose F is not an (AxS) or (Ax6). Then the assertion follows from the subsidiary I.H. using Lemma 8.

2. Suppose F is an (AxS) or (Ax6). Then n > 0 and the axiom is for a predicator f?JJ ..t such that

IOn - 1 I- Vx(U(x) -+ V(x» -+ Vx(d[U, x] -+ drY, x]).

By the I.H. we obtain

(1) PA~ I- Vx(U(x)-+ V(x»-+ Vx(d[U,x]-+ d[V,x]).

f?JJ ..t is the predicator

AzVY(Vx(d[Y, x] -+ Y(x» -+ Y(z»

of the system PA~+ 1. For each predicator P of the system IDn we have

PA~+l I- Vx(d[P, x] -+ P(x»-+ (VY(Vx(d[Y, x] -+ Y(x»-+ Y(t» -+ P(t».

Using a basic inference (S4) we obtain

(2) PA~+ 1 I- Vx(d[P, x] -+ P(x» -+ (f?JJ ..t(t) -+ P(t».

That is to say:

(3) PA~+1 I- F if Fis an (Ax6).

29. Proofs of Well-Ordering


PA~+ 1 I- 'Vx(&' d(X) --+ U(~» --+ (d[&' d' t] --+ d[U, t])

PA~+ 1 I- 'Vx(d[U, x] --+ U(x» --+ 'VX(&' d(X) --+ U(x».

From these two formulas we obtain

Using basic inferences (S2.1) and (S4) we obtain

o -- -PA,,+1 I- d[&' d' t] --+ &' d(t).

That is to say:

(4) PA~+1 I- F if Fis an (Ax5).

(3) and (4) prove the assertion.

Theorem 29.9. /fO<N<w, then: a) Formalized transfinite induction

Prog [TN' P] --+ 'VX(TN(X) --+ P(x»

over iJ(N) is deducible in the system PA~+ l'

291

b) The consistency of the formal system PAN+ 1 is provable by transfinite induction up to «w, UN)' 0) = O(Ow(UN + 1»0.

Proof a) follows from Theorems 29.7 and 29.8. b) Every normal deduction in the system PAN + 1 has an order IX E lJ(N + 1)

where IX«(W, UN)' Therefore dolX< «w, UN)' 0). The assertion therefore follows from the proofs of Theorems 27.8 and 27.9.

Remark. By W. Pohlers [2] the consistency of the formal systems IDN and PA~+1 is provable even by transfinite induction up to «(1, UN)' 0) = OSUN + 10 and formalized transfinite induction is deducible in these systems up to each ordinal term lk«I, UN)' 0).

Definition. Let TW+l be the deducible arithmetic predicate which holds if, and only if, fl E lJ(w+ 1).

Theorem 29.10. The ordinalOUwO is characteristic for the formal system PAw in the following sense:

a) Formalized transfinite induction up to n is provable in PA~for every ordinal term fl<OUwO that is to say, the formula

292 29. Proofs of Well-Ordering

Prog [T",+1> P] ~ P(n)

is deducible in PA~. b) The consistency of the formal system PA", is provable by transfinite induction

up to OQ",O.

Proof a) If ti < OQ",O there is a natural number N> 0 such that mE /}(N) for all m~ti. Then it follows from Theorem 29.9 a) that the formula

Prog [T",+ l' P] ~ P(n)

is deducible in PA~+1 and therefore in PA~ too. b) For each normal deduction H of the system PA", there is a natural number

N> 0 such that H is a normal deduction of the system PAN + 1.· Hence the assertion follows from Theorem 29.9 b) since O(Ow(QN + 1))0 < OQ",O.

Remarks. 1. By W. Buchholz [1] and [6] OQ",O is the order type of level 0 of the notation system I: of H. Pfeiffer [2]. This is at the same time the order type of the union of all invariant segments of the systems I:(N) (N E N) of K. Schiitte [10]. By H. Levitz [1], [2] and [3] this is simultaneously the order type of all the ordinal diagrams of finite order of G. Takeuti [2].

2. The proof of well-ordering given in the first part of this section for /}(r) can be formalized in a system with transfinitely iterated inductive definitions in the case where r = w + 1. But it cannot be formalized in any system IDn where n<w nor in GPA. This also holds for every proof of well-ordering up to the ordinal term « w2 , Q",), 0) of the system /}( w + 1). Transfinite induction up to this ordinal was used in §27 to prove the consistency of GP A.

3. It follows from work of W. Buchholz and W. Pohlers [5] that formalized transfinite induction is deducible in GPA up to each ordinal term

It can further be shown that this induction is deducible in PA up to each ordinal term ti < «0, Q", + 8 0), 0) = O(Q", . 80)0. It is possible to give consistency proofs for the systems GPA and PA with weaker transfinite inductions than were used in §§27 and 28. For by the result!> ofW. Buchholz [4] and W. Pohlers [3] if iXoPA and ocpA are the least ordinals up to which transfinite induction is sufficient for such consistency proofs then we have

Bibliography

Ackermann, W.:

[1] Zur Widerspruchsfreiheit der Zahlentheorie. Math. Ann. 117 (1940), 162-194. [2] Widerspruchsfreier Aufbau einer typenfreien Logik. (Erweitertes System). Math. Zeitschr. 55

(1952), 364-384. [3] Widerspruchsfreier Aufbau einer typenfreien Logik. II. Math. Zeitschr. 57 (1953), 155-166.

Aczel, P.:

[I] A new approach to the Bachmann method for describing countable ordinals. Mimeographed.

Bridge, J.:

[1] Some problems in mathematical logic. Systems of ordinal functions and ordinal notations. Thesis Oxford 1972.

[2] A simplification of the Bachmann method for generating large countable ordinals. Journal of . Symbolic Logic 40 (1975),171-185.

Buchholz, w.: [1] Rekursive Bezeichnungssysteme fUr Ordinalzahlen auf der Grundlage der Feferman-Aczelschen

Normalfunktionen 6 •. Dissertation Miinchen 1974. [2] Normalfunktionen und konstruktive Systeme von Ordinalzahlen. Proof Theory Symposion Kiel

1974. Springer Lecture Notes in Math. 500 (1975), 4-25. [3] Uber Teilsysteme von 9({g}). Archiv f. math. Logik u. Grundlagenf. 18 (1976),85-98. [4] Zur Beweisbarkeit der transfiniten Induktion in l1~-CA. Erscheint im Archiv f. math. Logik u.

Grundlagenf.

Buchholz, W. and Pohlers, w.: [5] Provable wellorderings of formal theories for transfinitely iterated inductive definitions. To appear

in Journal of Symbolic Logic.

Buchholz, W. und Schutte, K.:

[6] Die Beziehungen zwischen den Ordinalzahlsystemen E und 9(w). Archiv f. math. Logik u. Grundlagenf. 17 (1976),179-190.

Diller, J. und Schutte, K.:

[1] Simultane Rekursionen in der Theorie der Funktionale endlicher Typen. Archiv f. math. Logik u. Grundlagenf. 14 (1971),69-74.

Fe/erman, S.:

[I] Systems of predicative analysis. Journal of Symbolic Logic 29 (1964), 1-30. [2] Systems of predicative analysis, II: Representations of ordinals. Journal of Symbolic Logic 33

(1968), 193-200. [3] Autonomous transfinite progressions and the extent of predicative mathematics. Logic,

Methodology and Philosophy of Science III. Amsterdam 1968, 121-135.

294 Bibliography

[4] Hereditarely replete functionals over the ordinals. Intuitionism and Proof Theory. Amsterdam 1970,289-301. .

[5] Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. Intuitionism and Proof Theory. Amsterdam 1970, 303-326.

[6] Ordinals and functionals in proof theory. Proc. of the International Congress of Math. Nice 1970. Paris 1971, Vol. 1,229-233.

Gentzen. G.:

[I] Untersuchungen iiber das logische SchlieBen. Math. Zeitschr. 39 (1934),176-210,405-431. [2] Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann. 11Z (1936), 493-565. [3] Neue Fassung des Widerspruchsfreiheitsbeweises fiir die reine Zahlentheorie. Forschungen zur

Logik und zur Grundlegung det exakten Wissenschaften. Neue Folge 4 (1938),19-44. [4] Beweisbarkeit und Unbeweisbarkeit von Anfangsfallen der transfiniten Induktion in der reinen

Zahlentheorie. Math. Ann. 119 (1943), 149-161.

GOdel. K.:

[I] tiber formal unentscheidbare Siitze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys. 38 (1931), 173-198.

[2] tiber eine bisher noch nicht beniitzte Erweiterung des finiten Standpunktes. Dialectica 1~ (1958), 28(}-287.

Henkin. L.:

[1] Completeness in the theory of types. Journal of Symbolic Logic 14 (1949), 159-166.

Hilbert. D. und Bernays. P.:

[I] Grundlagen der Mathematik I, II, Die Grundlehren der math. Wissenschaften 40,50.2. Aufiage, Springer 1968, 1970. .

Howard. W.:

[I] Assignment of ordinals to terms for primitive recursive functionals offinite type. Intuitionism and Proof Theory. Amsterdam 1970,443-458.

[2] A system of abstract constructive ordinals. Journal of Symbolic Logic 37 (1972), 355-374.

Kreisel. G.:

[I] La predicativite. Bull. de Ia Soc. Math. de France 88 (1960), 371-391. [2] A survey of proof theory. Journal of Symbolic Logic 33 (1968), 321-388. [3] A survey of proof theory II. Proc. of the 2nd Scandinavian Logic Symposium. Amsterdam 1971,

109-170.

Levitz. H.:

[I] On a simplification of Take uti's ordinal diagrams of finite order. Zeitschr. f. math. Logik u. Grundl. d. Math. 15 (1969), 141-154.

[2] On the relationship between Takeuti's ordinal diagrams O(n) and Schiitte's system of ordinal notations 1:(n). Intuitionism and Proof Theory. Amsterdam 1970,377-405.

Levitz. H. and Schutte. K.:

[3] A characterization of Takeuti's ordinal diagrams of finite order. Archiv f. math. Logik u. Grundlagenf. 14 (1970), 75-97.

Lorenzen. P.:

[I] Aigebraische und logistische Untersuchungen iiber freie Verbiinde. Journal of Symbolic Logic 16 (1951), 81-106.

Pfeiffer. H.:

[I] Ausgezeichnete Foigen fUr gewisse Abschnitte der zweiten und weiterer Zahlenklassen. Dissertation Hannover 1964.

Bibliography 295

[2] Ein Bezeichnungssystem fUr Ordinalzahlen. Archiv f. math. Logik u. Grundlagenf. 12 (1969), 12-17. [3] Ein Bezeichnungssystem fUr OrdinaIzahlen. Archiv f. math. Logik u. Grundlagenf. 13 (1970),74-90. [4] Vergleich zweier Bezeichnungssysteme fUr Ordinalzahlen. Archiv f. math. Logik u. Grundlagenf.

15 (1972), 41-56. [5] Ober zwei Bezeichnungssysteme fUr Ordinalzahlen. Archiv f. math. Logik u. Grundlagenf. 16

(1974),23-36.

Pohlers, W.:

[I] Eine Grenze fUr die Herleitbarkeit der transfiniten Induktion in einem schwachen n:-Fragment der Analysis. Dissertation Miinchen 1973.

[2] An upper bound for the provability of transfinite induction in systems with N-times iterated inductive definitions. Proof Theory Symposion Kiel 1974. Springer Lecture Notes in Math. SOO (1975), 271-289.

[3] Ordinals connected with formal theories for transfinitely iterated inductive definitions. To appear in Journal of Symbolic Logic.

Prawitz, D.:

[1] Hauptsatz for higher order logic. Journal of Symbolic Logic, 33 (1968), 452-457. [2] Ideas and results in proof theory. Proc. of the 2nd Scandinavian Logic Symposium. Amsterdam

1971,235-307.

Schiitte, K.:

[1] Beweistheoretische Erfassung der unendlichen Induktion in der reinen Zahlentheorie. Math. Ann. 122 (1951),369-389.

[2] Beweistheoretische Untersuchung der verzweigten Analysis. Math. Ann. 124 (1952), 123-147. [3] Ein System des verkniipfenden SchlieBens. Archiv f. math. Logik u. Grundlagenf. 2 (1956),55-67. [4] Syntactical and semantical properties of simple type theory. Journal of Symbolic Logic 25 (1960),

305-326. [5] Beweistheorie. Die Grundlehren der math. Wissensch. 103. Springer 1960. [6} Logische Abgrenzungen des Transfiniten. Logik und Logikkalkiil. Freiburg, Miinster 1962,

105-114. . [7] Der Interpolationssatz der intuitionistischen Pradikatenlogik. Math. Ann. 148 (1962), 192-200. [8] Eine Grenze fiir die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik.

Archiv f. math. Logik u. Grundlagenf. 7 (1964), 45-60. [9] Predicative well-orderings. Formal Systems and Recursive Functions. Proc. of the 8th Logic

Colloquium Oxford 1963. Amsterdam 1965, 280--303. [10] Ein konstruktives System von Ordinalzahlen. Archiv f. math. Logik u. Grundlagenf. 11 (1969),

126-137; 12 (1969), 3-11. [11] Einfiihrung der Normalfunktionen 8. ohne Auswahlaxiom und ohne Regularitiitsbedingung.

Archiv f.-math. Logik u. Grundlagenf. 17 (1976), 171-178.

Tait, W. W.:

[1] Infinitely long terms of transfinite type. Formal Systems and Recursive Functions. Proc. of the 8th Logic Colloquium Oxford 1963. Amsterdam 1965, 176-185.

[2] A non constructive proof of Gentzen's Hauptsatz for second order predicate logic. Bull. Amer. Math. Soc. 72 (1966), 980--983.

[3] Normal derivability in c1assicallogic. The Syntax and Semantics of Infinitary Languages. Springer Lecture Notes in Math. 72 (1968),204-256.

Takahashi, M.:

[1] A proof of cut -elimination theorem in simple type theory. J. Math. Soc. Japan 19 (1967),399-410. [2] Simple type theory of Gentzen style with the inference of extensionality. Proc. Japan Acad. 44

(1968),43-45. [3) A system of simple type theory of Gentzen style with inference on extensionality, and the cut

elimination in it. Comment. universitatis Sancti Pauli, Tokyo 18 (1969), 129--147.

Takeuti, G.:

[1] On a generalized logic calculus. Japan J. Math. 23 (1953), 39--96. [2] Ordinal diagrams. J. Math. Soc. Japan 9 (1957), 386-394.

296 Bibliography

[3] Ordinal diagrams II. J. Math. Soc. Japan 12 (1960), 385-391. [4] Consistency proofs of subsystems of classical analysis. Ann. of Math. 86 (1967), 299-348. [5] Proof Theory. Studies in Logic 81. Amsterdam 1975.

Troelstra, A. S.:

[I] Metamathematical investigations of intuitionistic arithmetic and analysis. Springer Lecture Notes in Math. 344 (1973).

Subject Index

A.-abstraction 121 Aczel 4,224 addition in the system FT 120 - of ordinals 80, 223 - of ordinal terms 89,90 additive principal numbers 80, 224 arithmetic difference 123 atomic formulas 20, 57 autonomous ordinal terms 213,215,217 axioms of choice 150

bar-induction 254 basic deduction 256,271,275 - formula of the system FT 129 - inferences 14 Bemays-Godel set theory 74,221 Beth 36,55 Bridge 4 Brouwer 36 Buchhoa 4,225,230,282,292 bundle of formulas 256, 276

calculable arithmetic functions 167 Cantor normal form 81 cardinality of a set of ordinals 222 cardinals 222 characteristic term of a basic formula 130 closed formulas 115 - set of ordinals 223 coefficient sets 234, 235 combinators98 Compactness Theorem 34 complete induction inference 170, 254 Completeness Theorem 29, 35, 69 components of an ordinal 224 connectives 7 consistency 15 Consistency Theorem 17,28,35 consistent set of formulas 32 constant prime formulas 168 continuous ordering functions 78 Craig 51 critical ordinals and functions <p. 81, 82, 83

- parts of a formula 63 Curry 98 cut 37,170,200,254 Cut Elimination Theorem 203, 204 cut formula 200, 254 Cut rule 24, 62, 70

decidable arithmetic predicates 167 deducibility 14 D-chain (deduction chain) 29,33,63 deduction functions 248, 249 - part 263 - threat 262 degree of a basic inference 254 - ofa formula 41,252 - of an M-term 68 - of an ordinal of lJ(Q) 234 - of a term of T* 236 - of a type 106 D-form 40 !i)-formula 27 directly derivable inferences 16 distinguished part of a formula in classical

predicate calculus 29,62,63 - - ofa formula in a normal deduction 261,

277

eigenformula of an (S3.1)-inference 254 eigenvariable of an inference 20, 41, 60, 254 elementary analysis 177 - formulas 168 - number theory 176

,Embedding Theorem 175,273,290 end formula of a deduction 255, 275, 276 - part of a formula 40 - piece of a deduction 255, 256, 276 equality of terms of finite type 112 equivalent formulas 169 essentially less 247 explicit minimal parts of a formula in a normal

deduction 256, 276 Extensionality Theorem 173, 205

298

Feferman 4, 84,214,220,224,225 finite ordinals 76, 221, 222 - ordinal terms 90 finitely axiomatisable theories 54 - deducible formulas 176 formal system, CS of classical sentential calculus

17 - - CP of classical predicate calculus 19 - - IPI-IP3 of intuition is tic predicate calculus

36,38,40 - - cr of classical simple type theory 56 - - FT of functionals of finite type 113 - - PN of pure number theory 134 - - DA of Ll~~analysis 170 - - EN of elementary number theory 176 - - EA of elementary analysis 177 - - GPA of generalized n~-analysis 252 - - PA of n~-analysis 248 - - ID of n-fold iterated inductive definitions

286 • formalized transfinite induction 179,209,291 functionals of finite type 99

generalized inductive definitions 286 Gentzen 3,21,38, 147,220 Godel 2, 3, 62, 98, 134, 147

height of a type 58 Henkin 66 Heyting 36 Hilbert 2, 3, 26 Howard 3, 98, 105

identity functional 121 implicit minimal paris of a formula in a normal

deduction 256, 276 inaccessible ordinals and functions 8. 224, 225 inconsistent set of formulas 32 independence of premise ISO induction on the deduction 15 Induction Theorem 75, 128, 173,222 inductive definitions 8 initial formulas of a deduction 255, 271 Interpolation Theorem 51, 54 Interpretation Theorem 157,207,208 interpreting formulas ISS, 156 Inversion rules 22,61,201,272 isotonism 74 iterator 98

Konig's Lemma 30 Kreisel 55 Kripke 36

left parts of formulas and L-forms 39, 40 length of formulas and predicators 168, 198

Subject Index

- of a term of the set T 86 level of formulas and predicators 198 - of ordinals 224 Levitz 292 limit numbers 75,222 limiting number of a formal system 211 Lowenheim-Skolem Theorem 29 logical consequence 176

Markov's principle ISO maximal ex-critical ordinals and functions "'.

84,85 maximally ex-inaccessible ordinals 230, 232 metalogic, metamathematics 2 minimal parts ofa formula 12,20,169 model of classical predicate calculus 27 M-term, AI-formula, M-variant 65, 66 mUltiplication in the system FT 120

natural product of ordinal terms 105, 106 - sum of ordinal terms 105, 248 negative parts of formulas and N-forms 10, II,

20 nominal forms 1l,14 normal deduction 256, 275, 276 - form ofa term 104 - functions 78, 223 notation systems iJ(Cl) and iJ(T) for ordinals 237,

238 NP-forms II number terms 134 numerals 99,168 numerical terms 168

open terms 100 order of a deducible formula IS - of a formula in a normal deduction 257, 276 - of a normal deduction 258 - of a term. of finite type 108 ordering functions 77, 223 ordinal numbers 74,75 ordinals of the segment 0 74, 0* 221,

222 - of the set iJ(Cl) 234 ordinal terms. 213,215,217,236 - terms of the set OT 86, 87, OT* 236,

237 O-segment 76,77, O·-segment 223

partial valuation 63, 64 Peano axioms 167 permissible inferences IS IF-permissible formulas 199 Pfeiffer 292 PN-Lemma 27,66 Pohlers 257, 291, 292

Subject Index

positive parts of formulas and P-forms 10, II, 20

possible valuations 67 Prawiu 3,62,64,67 predecessor functional 122 predicators 168, 198 prime formulas of second order arithmetic 168 - terms of the system cr 58 primitive formulas 63, 268 - terms of finite type 98 principal inferences 200, 254, 270 - parts of axioms 170, 200, 253 - - of basic inferences 21,41,60, 200, 2~4 Principal Semantic Lemma 29,34,64 - Syntactic Lemma 29, 33, 64 principal terms of T 86, T* 236 progressive 75 progressiveness ~.[Pl 178, ~.[P"l 209,

Prog [', PI 287

quantifiers 19

rank of a cut 200 - of a deduction function 248, 249 - of a formula 199, 253 - of an inference 256, 276 - of a normal deduction 257 - of a term 58, 59 recursively comparable ordinal terms 237 recursor 125 reducibility degree ofa formula 18,29 - of a term of finite type 100 reducible and irreducible formulas 12, 13, 29,

62 - normal deductions 259,277 - parts of a formula 28, 62 reduction of terms of finite type 100, 101 regular terms of the system CT 58 regularity of a cardinal 222 Replacement rule 174, 201 right parts offormulas and R-forms 3.9, 40

Satisfiability Theorem 34 satisfiable set of formulas 32 Schwichtenberg 185 secondary part of an (-+ L)-inference 41 semantic 4 semantically equivalent sentential forms 8 semantic consequence 35 Semi-formal system, DA* of .1 I-analysis 174 - - EN* of elemllDtary number theory 177 - - EA· of elementary analysis 177 - - RA· of ramified analysis 197 - - PA* of nl-analysis 269 sentential form 8 - valuation 8

299

s-closed, s-complete, s-consistent, s-valid 16, 17 simple formulas of second order arithmetic 168 simultaneous primitive recursion 126 singular normal deduction 261,277 Soundness Theorem 66 specialization of a formula 273 strength of a cut 271, 272 - of a formula 270 strong formulas and predicators 252 strongly critical ordinals 83 structural inferences F j! G 20,40, 169 Structural rule 23,61,172,175,201,254 substitution inference 254 Substitution rule 22,42,61,114,136,172 subterm of an ordinal term 242 - of a term of the system cr 58 subterm-chain 58 successor of an ordinal 75, 222 - functional 98 - number 75 suitable cut of a normal deduction 261, 277 supremum 75, 76, 222 syntactic 4 - consequence 35 - equivalenceA~B 47, A£B 51 system of sets 65

Tait 62,204 Takahashi 3, 62, 64 Takeuti 4,62,221,252,257,292 terms of finite type 98 - of second order arithmetic 167, 168 - of the set T 86, T* 236 - of the system CT 57 tertium non datur 2, 3 total valuation 66 transfinite induction 75 Troelstra 151 truth functions ·7 - values 7 truth-value of a constant prime formula 169 types of functions 98 - of terms of the system cr 56

valid formulas 10, 11,27,66 valuations 27 value ofa numerical term 169

weak formulas and predicators 252 - inferences IS, 22, 23 weight of a formula 252 - of a normal deduction 257 well-ordered sets 73

Zermelo-Fraenkel set theory 74,221

Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete

Eine Auswahl

23. Pasch: Vorlesungen iiber neuere Geometrie 41. Steinitz: Vorlesungen iiber die Theorie der Polyeder 45. Alexandroff/Hopf: Topologie. Band I 46. Nevanlinna: Eindeutige analytische Funktionen 63. Eichler: Quadratische Formen und orthogonale Gruppen

102. Nevanlinna/Nevanlinna: Absolute Analysis 114. Mac Lane: Homology 123. Yosida: Functional Analysis 127. Hermes: Enumerability, Decidability, Computability 131. Hirzebruch: Topological Methods in Algebraic Geometry 135. Handbook for Automatic Computation. Vol. I/Part a: Rutishauser: Description of ALGOL 60 136. Greub: Multilinear Algebra 137. Handbook for Automatic Computation. Vol. IjPart b: Grau/Hill/Langmaack:

Translation of ALGOL 60 138. Hahn: Stability of Motion 139. Mathematische Hilfsmittel des Ingenieurs. I. Teil 140. Mathematische Hilfsmittel des Ingenieurs. 2. Teil 141. Mathematische Hilfsmittel des Ingenieurs. 3. Teil 142. Mathematische Hilfsmittel des Ingenieurs. 4. Teil 143. Schur/Grunsky: Vorlesungen iiber Invariantentheorie 144. Weil: Basic Number Theory 145. Butzer/Berens: Semi-Groups of Operators and Approximation f46. Treves: Locally Convex Spaces and Linear Partial Differential Equations 147. Lamotke: Semisimpliziale i!lgebraische Topologie 148. Chandrasekharan: Introduction to Analytic Number Theory 149. Sario/Oikawa: Capacity Functions 150. losifescu/Theodorescu: Random Processes and Learning lSI. Mandl: Analytical Treatment of One-dimensional Markov Processe~ 152. Hewitt/Ross: Abstract Harmonic Analysis. Vol. 2: Structure and Analysis for

Compact Groups. Analysis on Locally Compact Abelian Groups 153. Federer: Geometric Measure Theory 154. Singer: Bases in Banach Spaces I 155. Miiller: Foundations of the Mathematical Theory of Electromagnetic Waves 156. van der Waerden: Mathematical Statistics 157. Prohorov/Rozanov: Probability Theory. Basic Concepts. Limit Theorems. Random Processes 158. Constantinescu/Cornea: Potential Theory on Harmonic Spaces 159. Kothe: Topological Vector Spaces I 160. Agrest/Maksimov: Theory of Incomplete Cylindrical Functions and their Applications 161. Bhatia/Szego: Stability Theory of Dynamical Systems 162. Nevanlinna: Analytic Functions 163. Stoer/Witzgall: Convexity and Optimization in Finite Dimensions I 164. Sario/Nakai: Classification Theory of Riemann Surfaces 165. Mitrinovic/Vasic: Analytic Inequalities 166. Grothendieck/Dieudonne: Elements de Geometrie Aigebrique I 167. Chandrasekharan: Arithmetical Functions 168. Palamodov: Linear Differential Operators with Constant Coefficients 169. Rademacher: Topics in Analytic Number Theory 170. Lions: Optimal Control of Systems Governed by Partial Differential Equations 171. Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces 172. Biihlmann: Mathematical Methods in Risk Theory 173. Maeda/Maeda: Theory of Symmetric Lattices 174. Stiefel/Scheifele: Linear and Regular Celestial Mechanics. Perturbed Two-body

Motion-Numerical Methods-Canonical Theory

302

175. Larsen: An Introduction to the Theory of Multipliers 176. Grauert/Remmert: Analytische Stellenalgebren 177. Fliigge: Practical Quantum Mechanics I 178. Fliigge: PraCtical Quantum Mechanics II 179. Giraud: Cohomologie non abelienne 180. Landkof: Foundations of Modern Potential Theory 181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I 182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II 183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III 184. Rosenblatt: Markov Processes. Structure and Asymptotic Behavior 185. Rubinowicz: Sommerfeldsche Polynommethode 186. Handbook for Automatic Computation. Vol. 2. Wilkinson/Reinsch: Linear Algebra 187. Siegel/Moser: Lectures on Celestial Mechanics 188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I 189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II 190. Faith: Algebra: Rings, Modules, and Categories I 192. Mal'cev: Algebraic Systems 193. P6Iya/Szego: Problems and Theorems in Analysis I 194. Igusa: Theta Functions 195. Berberian: Baeu-Rings 196. Athreya/Ney: Branching Processes 197. Benz: Vorlesungen iiber Geometrie der Algebren 198. Gaal: Linear Analysis and Representation Theory 199. Nitsche: Vorlesungen iiber Minimalfiiichen 200. Dold: Lectures on Algebraic Topology 201. Beck: Continuous Flows in the Plane 202. Schmetterer: Introduction to Mathematical Statistics 203. Schoeneberg: Elliptic Modular Functions 204. Popov: Hyperstability of Control Systems 205. Nikol'skii: Approximation of Functions of Several Variables and Imbedding Theorems 206. Andre: Homologie des Algebres Commutatives 207. Donoghue: Monotone Matrix Functions and Analytic Continuation 208. Lacey: The Isometric Theory of Classical Banach Spaces 209. Ringel: Map Color Theorem 210. Gihman/Skorohod: The Theory of Stochastic Processes I 211. Comfort/Negrepontis: The Theory of Ultrafilters 212. Switzer: Algebraic Topology-Homotopy and Homology 213. Shafarevich: Basic Algebraic Geometry 214. van der Waerden: Group Theory and Quatum Mechanics 215. Schaefer: Banach Lattices and Positive Operators 216. P6Iya/Szego: Problems and Theorems in Analysis II 217. Stenstrom: Rings of Quotients 218. Gihman/Skorohod: The Theory of Stochastic Processes II 219. Duvaut/Lions: Inequalities in MeChanics and Physics 220. Kirillov: Elements of the Theory of Representations 221. Mumford: Algebraic Geometry I: Complex Projective Varieties 222. Lang: Introduction to Modular Forms 223. Bergh/Lofstrom: Interpolation Spaces. An IntroduCtion 224. GilbargfTrudinger: Elliptic Partial Differential Equations of Second Order 225. Schiitte: Proof Theory

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