Logical Frameworks for Truth and Abstraction (Studies in Logic and the Foundations of Mathematics, 135)

LOGICAL FRAMEWORKS FOR TRUTH AND ABSTRACTION An Axiomatic Study

STUDIES IN LOGIC AND THE

FOUNDATIONS

OF

VOLUME

MATHEMATICS 135

Honorary Editor: E SUPPES

Editors: S. ABRAMSKY, London S. ARTEMOV, Moscow J. BARWISE, Stanford H.J. KEISLER, Madison A.S. TROELSTRA, Amsterdam

ELSEVIER A M S T E R D A M 9L A U S A N N E 9N E W Y O R K 9O X F O R D 9S H A N N O N 9T O K Y O

LOGICAL FRAMEWORKS FOR TRUTH AND AB STRACTION An Axiomatic Study

Andrea CANTINI

Department of Philosophy Universityof Florence Florence, Italy

1996

ELSEVIER AMSTERDAM

9L A U S A N N E

9N E W Y O R K

9O X F O R D

9S H A N N O N

9T O K Y O

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 82306 9 9 1996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, EO. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the Publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-flee paper. Printed in The Netherlands

PREFACE

This book is concerned with logical systems, which are usually termed typefree or self-referential and emerge from the traditional discussion on logical and semantical paradoxes. We will consider non-set-theoretic frameworks, where forms of type-free abstraction and self-referential truth can consistently live together with an underlying theory of combinatory logic. However, this is not a book on paradoxes; nor we aim at a grand logic la Frege-Russell, inspired by a foundational program. We shall rather investigate type-free systems, in order to show that" (i) there are rich theories of self-application, involving both operations and truth, which can serve as foundations for property theory and formal semantics; (ii) these theories give new outlooks on classical topics, such as inductive definitions and predicative mathematics; (iii) they are promising as far as applications are concerned. This way of looking is justified by the history of the antinomies in our century. In spite of isolated foundational and philosophical traditions, the research arising from paradoxes has become progressively closer to the mainstream of mathematical logic and it has received substantial impulse during the last twenty years: a number of significant developments, techniques and results have been cropping up through the work of several logicians (see below for our main debts). Therefore a major aim of this book is to attempt a unifying view of relevant research in the field, by dwelling on connections with well-established logical knowledge and on applicable theories and concepts. However, the present work is far from being comprehensive. We do not treat illative combinatory logic (with the exception of a system of Ch.VI, investigated by Flagg and Myhill 1987), nor we deal with the BarwiseEtchemendy approach to self-reference via non-well-founded sets. Another significant direction, which is only touched upon in two sections of chapter XIII, is the systematic development of the general theory of semi-inductive definitions (in the sense of Herzberger, Gupta and others).

vI

Preface

The project started some years ago, when Prof. A. S. Troelstra kindly suggested an English translation of the author's monograph (Cantini 1983a) about theories of partial operations and classifications in the sense of Feferman (1974). The attempted translation soon shifted towards a thorough expanded revision of the old text, and eventually gave rise to an entirely new set of notes at the end of 1988. After a stop of almost two years, these notes were taken up again, fully rewritten and reorganized. The manuscript was submitted to the editor for final refereeing in October 1993. The content and the results of the present version are disjoint from the 1983 monograph; they partly overlap with the 1988 notes, except for a different choice of primitive notions and for the addition of Ch.VI, parts of Ch.XIII and the epilogue. Ch.VIII offers a development of topics, contained in the author's paper "Levels of Truth" (to appear in the Notre Dame Journal of Formal Logic, 1995): we thank the Editors for granting the permission of using parts of that paper in Ch.VIII of this book.

Acknowledgments. The present work owes a great deal to the writings of several logicians, and even if I tried hard to make a complete list of my debts in the text and in the reference list, I am sure that there are omissions: I apologize for them. As to the proper content of the book, pertaining to type-free abstraction and self-referential truth, I would like to underline my intellectual debt with the following papers (listed in alphabetical order): Aczel(1980), Feferman (1974), (1984), (1991), Fitch(1948), (1967), Friedman and Sheard (1987), Kripke (1975), Myhill (1984), Scott (1975). Profl W. Buchholz offered an invaluable help in correcting errors of any kind and in proposing technical improvements. I owe a special thank to him, also because the topics I dealt with were not touching his main research interests. I am grateful to Prof. S. Feferman and to Profi G. Js for keeping me informed over the years about their own research on type-free systems and proof theory, and for important advice. J~iger's Ph.D. student, T. Strahm made useful critical comments on the first chapter. Dr. R. Giuntini and Dr. P. Minari undertook the final proof-reading of chapters I-VIII and XII-XIV; I warmly thank them for a host of useful remarks and corrections. I am deeply indebted to Dr. A. P. Tonarelli for proof-reading the remaining chapters and for eagle-eyed assistance in the unrewarding task of preparing the final manuscript.

Preface

vii

Of course, I must stress that I am fully responsible for all errors, to be found in the whole work. I am grateful to the Alexander-von-Humboldt Stiftung (Germany) for granting me a "Wiederaufnahme" of a research fellowship at the LudwigMaximilians-Universit~it M/inchen in Sommer Semester 1991, when the present work was at a difficult stage. Partial support to the present project was granted by the Italian National Research Council (CNR)-and the Italian Ministry for University, Scientific Research and Technology (MURST). Last but not least, this work is dedicated to my children Giulia and Francesco.

Firenze, April 1995

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CONTENTS

Preface Contents

IX

Introduction PART A: COMBINATORS AND TRUTH Introducing operations The basic language 2. Operations I: general facts Operations Ih elementary recursion theory 3. 4A. The Church-Rosser theorem 4B. Term models The graph model 5. An effective version of the extensional model D co 6. Appendix

13 14 15 18 22 26 28 34 39

Extending operations with reflective truth 7. Extending combinatory algebras with truth 8. The theory of operations and reflective truth: simple consequences 9A. Type-free abstraction, predicates and classes 9B. Operations on predicates and classes 10A. The fixed point theorem for predicates 10B. Applications to semantScs and recursion theory 11. Non-extensionality Appendix I: a property theoretic definition of the fixed point operator for predicates Appendix Ih on the explicit abstraction theorem Appendix III: independence of truth predicates from the encoding of logical operators

43 45

o

II

11

51 55 59 63 68 73 76 77 80

Contents

x

PART B: TRUTH AND RECURSION THEORY III

IV

V

Inductive models and definability theory 12. Inductive models and the induction theorem 13. The envelope of an inductive model 14. The uniform ordinal comparison theorem for inductive models 15. Applications of the uniform ordinal comparison theorem

VII

85 86 88 91 97

Type-free abstraction with approximation operator 16. Approximating properties by classes 17. The approximation theorem for extensional operations and the fixed point theorem for monotone operations 18. Topology displayed: basic definitions 19. The representation theorem for explicitly CL-continuous operators Appendix: alternative proofs

103 104

Type-free abstraction, choice and sets 20. Choice principles and the distinction between operations and functions 21. Admissible hulls: elementary facts 22. A model of admissible set theory 23. The boundedness theorem

125

PART C: SELECTED TOPICS VI

83

109 113 117 122

126 131 137 144 149

Levels of implication and intensional logical equivalence 24. Myhill's levels of implication 25. Formal deducibility based on levels of implication and its proof-theoretic strength 26. Introducing an intensional equivalence relation 27. The infinitary reduction relation :=~ 28. The Church-Rosser theorem for ==~ 29. A model of type-free logic based on intensional equivalence

151 152

On the global structure of models for reflective truth 30. The lattice of fixed point models for the neutral minimal theory 31. The sublattice of intrinsic fixed point models and the cardinality theorem

177

158 162 165 169 174

179 186

Contents 32. 33. 34. 35.

Variations on the encoding technique: non-modularity and other oddities A model for an impredicative extension of reflective truth On Kripke's classification of self-referential sentences On the consistency of coinduction principles Appendix: a variant to the basic operator F and the restriction axiom

XI

192 198 203 207 209

PART D: LEVELS OF TRUTH AND PROOF THEORY

213

VIII Levels of reflective truth 36. A language and axioms for reflective truth with levels 37. Simple consequences 38. Universes and the Weyl extended iteration principle 39. A recursion-theoretic model 40. Levels of truth and predicatively reducible subsystems of second-order arithmetic 41. Consistency of a reducibility principle for classes 42. Levels of truth and impredicative subsystems of second-order arithmetic Appendix: on projectibility and stronger reflection

215 218 220 225 230

IX

Levels of truth and predicative well-orderings 43. On well-orderings 44. Ramified hierarchies 45. Predicative well-orderings I 46. Predicative well-orderings II

257 258 261 269 277

X

Reducing reflective truth with levels to finitely iterated reflective truth 47. A sequent calculus STLR for a theory of reflective truth with levels 48. Basic properties of STLR 49A. Elimination of the full level induction schema 49B. Elimination of unbounded level quantifiers 50. The infinitary sequent calculus I T ~ of n-iterated reflective truth 51. Embedding STLR n into I T ~

XI

Proof-theoretic investigation of finitely iterated reflective truth 52. The ramified system RS n 53. Cut elimination 54. Some derivable sequents of RS n

238 244 248 253

285 286 291 293 297 303 305 311 312 316 320

XII

Contents 55. 56. 57. 58.

Embedding ITn~ into RS n The upper bound theorem for I T ~ Upper bound theorems for TLR and its subsystems Conclusion: the conservation theorems Appendix: primitive recursive cut elimination for RS n

PART E: ALTERNATIVE VIEWS XII

Non-reductive systems for type-free abstraction and truth 59. The core system V F - and transfinite induction 60. Supervaluation models of V F 61. An abstract sequent calculus and truth 62. Cut elimination and related properties 63. A provability interpretation and the upper bound theorem 64. Reconciling supervaluation models with provability interpretation

XIII The 65. 66. 67. 68. 69.

variety of non-reductive approaches An inconsistency On a truth theory of Friedman and Sheard Fitch's models Introducing semi-inductive definitions Semi-inductive models for reflective truth

324 327 329 335 338 349 351 352 357 358 364 369 375 379 380 383 386 390 394

XIV Epilogue: applications and perspectives 70A. A logical theory of constructions: informal motivations 70B. A logical theory of constructions: basic syntax 71. Axioms for the computation relations 72. Extending the logical theory of constructions with higher reflection 73. Proof-theoretic reduction 74. Perspectives: related work in Artificial Intelligence and Theoretical Linguistics 75. Sense and denotation as algorithm and value: subsuming theories of reflective truth under abstract recursion theory

401 402 403 407

Bibliography

425

Index

441

List of Symbols

453

411 416 419 422

INTRODUCTION "There never were set-theoretic paradoxes, but the property-theoretic paradoxes are still unresolved" (K. Gbdel, as reported by J. Myhill 1984). "... the theory of types brings in a new idea for the solution of the paradoxes, especially suited to their intensional form. It consists in blaming the paradoxes not on the axioms that every propositional function defines a concept or a class, but on the assumption that every concept gives a meaningful proposition, if asserted for any arbitrary object or objects as arguments" (K. G6del 1944)

1. Informal ideas. The starting point of our investigation is the idea that the notions of predicate application and property are susceptible of independent study; in particular, these intuitive notions should be kept distinct from their counterparts of set-theoretic membership and set, as it is readily seen through a brief comparison. According to the iterative conception, a set is always a collection of mathematical entities of a given type (possibly, sets of lower rank); thus it has its being in its members, and equality among sets is ruled by the extensionality principle. Sets are conceived as completed totalities, generated by language independent operations and iterations thereof. The membership relation is a standard mathematical relation: this means that a C b is a well-defined proposition, whenever a and b are sets. Moreover, if we reflect upon the intuitive picture of the cumulative hierarchy, we come to know that C is well-founded and does not allow self-application. By contrast, a property is an abstract object, which is grounded in a concept, i.e. a function, not in the objects which fall under it (Frege 1984, p.199); it has no a priori bound on its extension, and it usually depends on some sort of explicit or implicit finite specification. Properties satisfy the so-called unrestricted abstraction or comprehension principle (AP): every condition A(x) determines a property {x:A}, which applies to all and only those things of which A(x) holds true. Of course, on the face of the well-known paradoxes, A P introduces elemcnts which are open to dispute and to multiform solutions; for instance, as GSdel's citation suggests, the predication r e l a t i o n - henceforth 7/- cannot be always meaningful, and therefore the laws of standard (classical) logic cannot be valid.

2

Introduction

The present approach, to be developed in various forms in this book, tries to keep the regimentation for predication and abstraction at a minimum; we maintain that {x" A} is an individual term and that r/applies to statements possibly involving 7/ itself. Thus we are looking for flexible, type-free theories of predication. More specifically, we are influenced by the tradition of illative combinatory logic in the sense of Curry and Fitch, by the work of Feferman (1975) on partial classifications and of Aczel (1980) on Frege structures. The inspiring idea is that properties and predication can be adequately explained in terms of the primitive notions of function and truth. As to the notion of function, we cannot expect to deal with functions in set-theoretic sense. In fact properties, given in intension, may apply to anything in a given realm, without type restrictions; and the same must hold of the functions underlying the properties themselves. Thus we are driven to understand functions essentially as rules of constructions (or, in short, operations) in the sense of combinatory logic. In contrast to the set-theoretic conception, operations are prior to their graphs and have no a priori bound on their domain; in particular, they do support non-trivial forms of self-application. On this view, it is natural to identify properties

with object-correlates of functions, and to reduce the abstraction operation to familiar )~-abstraction; formally, {x:A} simply becomes a h-term of the form )~x[a], where [A] is a term of combinatory logic, canonically representing the function defined by the condition A (of any given language). The second point concerns the reduction of predication to a primitive notion of reflective (or self-applicable) truth. Indeed, the expression

yq{x : A} is analyzed as: " the result of applying the function represented by {x:A} to the argument y turns out to be true". Therefore, if we let T stand for the truth predicate, yq{z: A} is defined as T({x: A}y) (with juxtaposition of {x: A} and y as application), and the abstraction principle AP becomes obviously derivable from the basic law of h-abstraction (i.e. we convert {x: A}y to the term [A[x := y]], the result of replacing x with y in [A]). Of course, these preliminary considerations do not solve the main problem of specifying the basic features of the truth predicate T. Nevertheless, they direct our attention towards the study of simple mathematical objects, namely expansions of combinatory algebras by reasonably closed truth sets. The typical structure (essentially) consists of a pair

where (i) 3t~ is a combinatory algebra, i.e. a model of Curry's combinatory

Introduction

3

logic; (ii) ff is a subset of M ( - t h e domain of 31,), which assigns a semantical structure to Jtt~. These expansions are uniformly described by means of operators F from the power-set of M into itself, acting as abstract valuation schemata. Informally, if X C_ M, F(X) represents the set of "truths" we come to know by means of the semantic rules of F on the basis of X. A central role in this book is played by an operator F, which essentially embodies Kleene's three valued non-strict interpretation of logical constants. In general, if F is monotone and reflects a cumulative conception of knowledge, the natural candidates for o-j-will be those subsets of M, that cannot be further extended with new information by means of F, i.e. the so-called fixed points of F, satisfying F(X) - X. Among these sets, a special role will be played by the minimal ones: they are technically the most interesting objects for the recursion-theoretic and proof-theoretic investigations. Conceptually, they reflect the idea that abstraction is not the mere description of an independent logical realm, but rather a process with its own logic implicit in F. In order to provide a few intuitions behind the construction of the first part of the book, it may be suggestive to regard 31~ as an abstract syntax, in which formal languages can be processed and defined. In particular, elements of M may be thought of as symbolic expressions, to be combined and identified according to the operations and laws of combinatory logic. M will typically include (notations for) natural numbers or any other chosen ground type, but also and most important, objects representing functions. The objects associated to computable functions can be seen as (functional) programs, implementing effective algorithms. On the other hand, still pursuing the computational analogy, properties-as representatives of (generally non-computable) propositional functions-can be considered as programs implementing a sort of generalized algorithms. While application of an effective algorithm to an input produces a computation, possibly converging to a value, a property { x ' P ( x ) } is ultimately applied to an object, in order to produce a verification that the object itself satisfies the given condition, according to the rules specified by the truth set ~. We wish to conclude by raising three conceptual points. First of all, the notion of truth T is not understood as a formalized truth predicate in the usual metamathematical sense: T classifies the objects of a combinatory algebra, and not an inductively defined collection of sentences ! In this sense, T, like set-theoretic membership, does not depend upon a specific language. As it should be clear from the sketched schema of interpretation ~ + , the predicate T is a primitive concept, which is prior to the specification of any formalism and is the source of the abstract notion of proposition. The underlying philosophy is that there are certain objects in our universe AI~,

4

Introduction

which carry information and can be called propositions; they can be partitioned into atomic or complex. Atomic propositions are simply grasped and reflect implicit (synthetic) knowledge, to be accepted as given. On the other side, complex propositions correspond to some sort of construction via logical operators; thus they require a(n analytic) process, in order to be understood (think of the search for verification), and they are controlled by the truth predicate T. As a second point, we like to stress the importance of having operations acting on classifications. Indeed, the fact that operations and classifications live together has the consequence of a symmetry, lacking to set theory: not only we can classify operations, but we can operate on classifications. So we can treat classifications, which depend on parameters, as operations; this is generally impossible in set theory. It follows that many constructions and statements acquire an "explicit character" and greater uniformity. A final comment is left for the choice of non-extensional basic notions. In general, even if we make use of intensional data (like definitions or enumerations), we never appeal to specific features of them, and thus we obtain results with an intrinsic character. Moreover, we find that the nonextensional language helps to avoid "strong logical principles" and to carry out proofs in rather weak systems (just as remarked in Kreisel 1971, p. 170); it often permits uniform and explicit statements of the results obtained, which do not obscure the appreciation of proper extensional aspects. On the contrary, non-extensional and extensional features are free to interact in a unified framework. As it will be clarified by the introduction of the approximation structure in chapter III and its applications in the subsequent chapters, the essential interplay of these aspects leads to rather smooth generalizations of the Myhill-Shepherdson theorem (w to the appreciation of extensional choice principles (w and to a satisfactory "internal" treatment of inductive definitions (boundedness and covering; w 2. Organization and contents. As we previously explained, the starting point of the book is the need for an independent logical approach to the notions of predicate application, property, abstraction, truth. The arrangement of the material reflects the increasing logical complexity of the truth notions that are met in the text. The different proposals, though generally motivated by model-theoretic constructions, are developed in axiomatic style. This is mainly because we wish to emphasize the connections with standard concepts of mathematical logic and deductive systems for (substantial parts of) mathematics. Proof-theoretic considerations and conservative extension results play an important role in classifying the various systems: very loosely, we tend to stress the importance of frameworks not stronger than Peano arithmetic and

Introduction

5

to distinguish predicative from impredicative systems. We also underline that type-free systems should not be opposed to type theories; we regard the former as a sort of generalized type assignment systems, in which types are left implicit and emerge from the theory itself. More concretely, the book is divided into five parts, which group together relatively homogeneous topics. The read thread can be described as follows. By and large, the first three parts form a sort of independent essay on a first-order theory of reflective truth over combinatory logic, whose truth axioms essentially stem from Fitch's extended basic logic (Fitch 1948) through Scott (1975) and Aczel (1977). The notion of reflective truth explicitly refers to Feferman (1991). After the general results of Part A, the theory is motivated and enriched by means of recursion-theoretic investigations (part B), by showing its unifying power and studying its semantics (part C). Parts D and E explore alternative routes. Part D deepens the intuitions underlying the systems of parts A-C by use of prooftheoretic techniques and by relativizing the concept of truth. Part E is experimental in character and scans over a variety of approaches, which are still subject of investigation. To give the reader a closer idea of what is in the book, we shall survey the content of the single chapters. A more detailed account can be found in the introductory section to each chapter. Part A: it offers a general introduction to the basic notions of operation and reflective truth. The basic aim is to illustrate, both axiomatically and semantically, a consistent notion of type-free logical structure, which will be fundamental in the whole book. The opening chapter contains the necessary preliminaries on (expanded) combinatory logic, which is here taken as the core of a classical first-order theory of operations OP. There is an introduction to concrete models of OP, as they form the ground structures in the entire book. In the second chapter, we inductively expand combinatory algebras with a notion of self-referential truth, which naturally generalizes the familiar Tarskian semantical clauses, in order to cope with a situation of partiality. The given expansions only depend on the isomorphism type of the underlying combinatory algebras. By inspection of the model-theoretic construction, we are led to a minimal axiomatic first-order system MF-, which contains a version of the Kripke-Feferman axioms for reflective truth and yields a theory of partial and total properties ( = classes henceforth), satisfying natural closure conditions. For instance, classes are provably closed under Feferman's join and elementary comprehension principles; moreover, MF- is provably closed under inductive definitions (though not capable of showing the corresponding induction schemata). We also consider

6

Introduction

extensions of MF- with various number-theoretic induction principles. Part B: we show that there is a two-sided link between generalized recursion theory and languages with operations and self-referential truth. Not only inductive definitions are crucial for building up models of self-referential languages, but these languages offer smooth formulations of non-trivial definability results. In chapter III we prove that classes (properties) in the inductive model over a given combinatory algebra a~ exactly define the hyperelementary (inductive) subsets of dtt, in the sense of Moschovakis (1974). The recursiontheoretic approach suggests to extend the minimal system by simple approximation conditions on properties. The new axioms, together with MF-, a powerful generalized induction schema GID and number-theoretic induction for classes, form an axiomatic system PWc+GID , which is still conservative (actually proof-theoretically reducible to) over the theory of operations and hence over Peano arithmetic. In chapter IV we show that PWc+GID proves a number of interesting consequences (separation and reduction principles) and, above all, an analog of the Myhill-Shepherdson theorem for operations which are y-extensional (i.e. extensional with respect to the predication relation). The results can be restated in topological terms via a natural generalization of the positive information topology. In chapter V, we produce models for admissible set theory and a boundedness theorem for inductive sets, again provably in PWc+GID. Part C: it is a natural complement to the previous parts. In chapter VI, the reader will find two alternative type-free logics. The first system, due to Myhill (1972, 1980), relies on a logic with levels of implication. The second system, inspired by Aczel-Feferman (1980), offers a type-free logic with a definitional equivalence relation on formulas, which is inspired by conversion in combinatory logic. Both systems are formally interpreted in the theory PWc+GID of chapters IV-V. Chapter VII offers a general outlook on the global structure FIX(art,) of fixed point models of N M F - ( = M F - without a consistency axiom) over arbitrary combinatory algebras art,. We prove that FIX(.Jt) only depends on the isomorphism type of art, and that the set of sentences A such that TA holds in every structure of FIX(..~), for arbitrary d~, is axiomatizable. It is shown that FIX(art) is a very rich and intricate non-distributive complete lattice; a few applications to consistency results and to formal semantics are thereby outlined (see connection with Kripke 1975). Part D: it focuses on proof theory and the foundations of mathematics. We investigate a type-free logic TLR, which is able to internalize to a certain

Introduction

7

extent quantification on classes and negative semantic information. The intuitive idea is that truth is the (direct) limit of local self-referential truth predicates, which are related one another by a directed pre-order of levels. Formally, we present TLR and its variants in chapter VIII. Among its consequences, it is possible to introduce a notion of "mathematical universe" with nice closure properties and interpret non-trivial subsystems of second-order arithmetic (ranging from versions of predicative analysis, like Friedman's ATR0, to the so-called A12-CA). In chapter IX we develop the prerequisites for a proof-theoretic analysis of TLR: in particular, we describe a well-ordering proof of the so-called Feferman-Sch/itte ordinal. Chapter X proves that the theory of truth with levels is proof-theoretically reducible to (infinitary) theories of finitely iterated self-referential truth ITS; on the other hand, each I T ~ is shown to be reducible to fragments of predicative analysis in chapter XI. The methods used include cut elimination for ramified systems in w-logic and asymmetrical interpretations d la Girard. Part E: we are concerned with logics of truth and type-free abstraction, which are based upon non-reductive, non-truth functional semantical valuation schemata. In contrast to the reductive schema underlying the semantics of chapter II, we study systems which are well-behaved with respect to logical consequence (e.g. a tautology is always classified as true; this does not work under a partial semantics d la Kleene). Chapter XII investigates a minimal system VF endowed with a simple supervaluation monotone semantics; VF naturally justifies principles of generalized inductive definitions (in contrast to what happens with the theories of parts A-C, it yields a model of the theory of elementary inductive definitions ID1). We also develop an alternative interpretation for VF by means of proof-theoretic infinitary methods. Chapter XIII addresses the problem of extending the logic of truth, as codified in VF. We discuss a refinement of supervaluation methods; but the new point is the introduction of semi-inductive definitions (in the sense of Herzberger 1982) and the application of the related notion of stable truth. We also consider consistent though w-inconsistent logics of truth, due to Friedman, Sheard and Mc Gee. The epilogue (chapter XIV) discusses prospective applications of typefree systems, as they result from the literature. In particular, we consider a logical theory of constructions, that has been investigated in Theoretical Computer Science and is strictly linked with the theories of part D. We conclude with a short survey of applications in other fields.

Introduction

8

3. How to use the book. The interdependence of the chapters is roughly indicated in the diagram below:

I II

1 III IV

VII

V VIII

~

XlI

IX

VI

1

x

1

1

XI

~

XlII

XIV Certain parts of the book, once suitably combined, offer a non-conventional approach to: 1) generalized recursion theory and inductive definability (part A + part B); 2) predicative mathematics and subsystems of analysis (part A + part B + + part D). If we disregard the recursion-theoretic and proof-theoretic parts, the book can serve as an introduction to" 3) formal semantics (part A + III + part C + VIII (w167 36-39) 4- part E). If the reader has in mind possible connections with logics for Artificial Intelligence, Theoretical Computer Science or semantics for natural languages, the suggestion 3) can be profitably modified to: 4) part A + part B + VIII (w167 36-39) + part E. Some chapters have appendices, containing additional details for proofs or suggestions for alternative routes: they can be always skipped without prejudice of understanding the later developments. 4. Prerequisites. The text is intended for readers who are familiar with the topics usually covered in an advanced undergraduate or basic graduate logic course. Thus we assume acquaintance with the elements of first-order logic

Introduction

9

and model theory, recursion theory, set theory and proof theory, as they are developed in good standard textbooks, or in the corresponding chapters of the Handbook of Mathematical Logic (Barwise 1977). In particular, it is useful to have a preliminary knowledge of the basic facts of hyperarithmetic and inductive definability (see Aczel 1977a, Moschovakis 1974). For the proof theory of Chapters VIII-XI, a previous exposure to sequent calculi and w-logic would be helpful (e.g. see Schwichtenberg 1977 or the textbooks of Takeuti 1975, Schfitte 1977, Girard 1987, Pohlers 1989). The simple topological notions of Ch. IV can be obtained from any standard textbook in general topology. Ch. VII presupposes a few elementary facts about partially ordered sets and lattices, usually met in logic courses (consider the classical reference of Birkhoff 1967). In Ch. VIII we hinge upon some advanced results of admissible set theory, to be found in Barwise (1975), Hinman (1977); however, the basic definitions and results are briefly recalled there. 5. General notations and conventions. A number of notations are adopted in the whole text. We summarize them below. 5.1. Internal and bibliographical references. The book is structured in five parts from A to E; each part is subdivided into chapters; the chapters are organized in sections, which are numbered in progressive order. Within each section, each specific item (subsection, definition, remark, axiom, rule, theorem, lemma or corollary) is usually assigned a pair "m.n" of numbers: "m.n" refers to the nth item of the ruth section. Sometimes, for finer classifications and reference, we allow the use of three (and exceptionally four) numbers (e.g. 37.4.1 locates the first sub-item of the 4th-item of section 37). In some cases, we specify the class, which the referred item belongs to (e.g. we may speak of theorem 3.2 or definition 34.5). References to publications are given by means of the author's name followed by the year of publication, possibly followed by a letter in the case of more publications by the same author in the same year. 5.2. Definitions. := is used as the definition symbol (definiendum on the left of : = , definiens on the right), while - stands for literal identity, unless it is specified otherwise. 5.3. Variables and substitution. We adopt the standard notions of free and bound variable; FV(E) is the set of free variables of the expression E. E[x := t] denotes the substitution of x with t in E. E(E') means that E' possibly occurs as a subexpression of E. 5.4. Logical Symbols. As usual we use V, 3, -1, ~ , A, V, ~ . For bracketing, we adopt the usual conventions; V, 3, -~ bind stronger than the other symbols, while A, V bind more than ---, and ~ . To enhance

Introduction

10

readability, dots may be used instead of brackets as separating symbols. A A B.---,C, A---,.B V C, 3x.A stand for (A A B)---,C, A ~ ( B V C) 3xA (respectively); ~x.ts shortens ~x(ts), etc... Sometimes, we make use of bounded quantifiers as abbreviations: if R : = r / , E, VxRa.A, 3xRa.A shorten Vx(xRa~A), 3x(xRa A A). If bounded quantifiers are iterated, we write: (VxRa)(VyRb)(...), (VxRa)(3yRb)(...), or even VxRa.VyRb.(...), VxRa.3yRb.(...), for the proper Vx(xRa---,Vy(yRb...)), Vx3y(xRa A yRb...) (respectively). We shorten logical equivalence on the metalevel (i.e. "if and only if") by the standard "iff" . "3!x" stands for "there is exactly one x". Sometimes, we adopt :=~ as implication on the metalevel. 5.5. Logical Complexity. The logical complexity of any given formula A is the number of distinct occurrences of logical symbols in A. 5.6. Set-theoretic symbols. We use the standard E , ~ (negation of E ), w (the set of natural infinite ordinal), 0, U, A, ~P(X) (power set of (Cartesian product), f" X-~Y (to be read as Y"), cz (characteristic function for the set Z).

notations: numbers, but also the first

X), X - Y ,

C, C_, D, D_, •

"f is a function from X to If k,m E w,

[k, m] "- {i E w " k _< i _< m}; ( k , m ] ' - { i e w ' k < i _ < m } ; (k,m) . - {i

k < i < m}; [ k , m ) . - {i

k < i < m}.

{ x : . . . } is the set of objects satisfying the condition (...); {al,... , an} is the set containing exactly the elements a l , . . . , a n. (...) denotes set-theoretic ntuple operation, unless otherwise specified. We warn the reader that set-theoretic symbols will be sometimes adopted as abbreviations for corresponding non-extensional operations on properties and predicates. But possible ambiguities will be spared by the context. The arithmetical symbols are the standard ones. 5.7. Provability and standard Tarskian semantics. ~ I = A stands for "A holds in the structure Eft,". S F A means that A is derivable from S by means of classical logic (unless otherwise specified). 5.8. Inductive proofs. We often carry out proofs by induction (either in the metatheory or within axiomatic theories). As a rule, we adopt the acronym IH as a shortening for "induction hypothesis".

PART A

COMBINATORS AND TRUTH

~r v a s t & "V r ~ v a l r o p ~ v " M//~'~ #c7' c~lrrls: cvL ")'~p, ~ #a~c&pLc, carL, '" ~1 #cT"Larrl ~a't 7rpJorrI. 7rcp'~ ~/&p avrrlv a v r o v riTv apxrlv o ~ a rvTx~vet" (Plato, Soph.238a)

This Page Intentionally Left Blank

CHAPTER 1

INTRODUCING OPERATIONS w w w w w w w

The basic language Operations I: general facts Operations II: elementary recursion theory The Church-Rosser theorem Term models The graph model An effective version of the extensional model D oo Appendix

This chapter contains an elementary introduction to combinatory logic. The topic is highly developed, but the chapter has quite a limited aim: that of yielding all the necessary prerequisites and making the book self-contained. According to the informal ideas outlined in the general introduction, we aim at investigating an axiomatic notion of abstract logical system, whose ground structure (the abstract syntax) is a combinatory algebra, extended with suitable built-in operations and with a primitive notion N of natural number. The choice of N is largely a matter of convenience and tradition; the basic constructions do not depend on the initial stock of built-in predicates and operations. The central aim of this chapter is to clarify what we understand by ground structure. We begin in the axiomatic style and we describe a formal system OP for a type-free theory of operations; we then outline three basic semantic constructions. We underline that the basic constructions can be carried out in OP itself. After the description of the formal language (w we define OP and we discuss its general features (w closure under /?-conversion, fixed point theorem, relation with )~-calculus), while w reviews some basic facts on recursion theory. We then present the term models of OP, which are based on the fundamental Church-Rosser theorem (w In w we give an elementary description of the Plotkin-Scott graph model Pw, together with its recursive submodel R E and Engeler's construction. Finally, following an elegant procedure, due to Scott (1976, 1980), we show how to isolate an extensional submodel D oo of RE.

Introducing Operations

14

[Ch.1

w The basic language We describe an axiomatic theory of operations OP, which is a first-order extension of pure combinatory logic by simple number-theoretic notions. OP is proof-theoretically equivalent to PA, the elementary system of Peano arithmetic, s it will constitute the basis of all systems to be investigated in this book. The basic language 2, contains: (i) countably many individual variables Xl, x2, x3, ... ; (ii) the logical constants -1, A, V; (iii) the individual constants K (constant function combinator), S (composition combinator), SUC (successor), P R E D (predecessor), P A I R (ordered pair operation), L E F T (left projection), R I G H T (right projection), 0 (zero), D (definition by cases on numbers); (iv) the binary function symbol Ap (application operation) and the predicate symbols N (natural numbers), T (truth), = (equality). Terms are inductively defined from variables and constants via application of Ap. We use x, y, z, u, v, w, f, g as metavariables; while t, t', t ' , s, s ~, r, r ~, etc., are metavariables for terms. We write (ts) instead of Ap(t,s), and outer brackets are usually omitted, while the missing ones are restored by associating to the left; for instance, xyz stands for ((xy)z). We adopt familiar shorthands for special terms: t + 1 : - SUCt ( - the successor of t); (t,s):-- P A I R t s ( = the ordered pair composed by t and s); (t)i := L E F T t ( - t h e left projection of t ) a n d (t)2 : - R I G H T t ( = the right projection of t). Formulas are inductively generated by means of the logical operations from atomic formulas (atoms, in short) of the form t = s, Nt, Tt. A, B, C are syntactical variables for formulas of 2,. As to the syntactical notions of free and bound variable, substitution, etc., we follow the standard conventions and terminology (Shoenfield 1967). In particular, if E is an expression (term or formula), E(x) means that x may occur free in E, while E[x := t] stands for the result of substituting t for the free occurrences of x (provided t is substitutable for x in E). FV(E) is the set of free variables of the expression E; x E FV(E) means that "x occurs free in E~

9

The remaining logical symbols are defined classically:

3xA := -~Vx~A; A V B := -~(-~A A -~B); A - , B := ~A V B ; A + B := (A---,B) A (B---,A). We stick to the usual convention that --1 and quantifiers bind more than the remaining connectives, while A, V bind more than --, and ~-,; sometimes dots are used in place of parentheses (see w of the introduction).

Basic language

1.1]

15

As usual, a numeral is any term obtained from the constant zero by means of a finite number of successor applications; if n E w (w - the set of natural numbers), ~ stands for the n-th numeral, i.e. the term built up from 0 with n applications of S U C . We now recall the standard definition of A-abstraction in combinatory logic. 1.1. D E F I N I T I O N . If t is an arbitrary term of s induction on the notion of s (i) Ax.x "- S K K ; (ii) Ax.t : - K t if x it FV(t); (iii) A x . ( t s ) ' - S ( A x . t ) ( A x . s ) , if x E FV(ts)

A x . t is introduced by

Of course Ax.t has exactly the same free variables as t, minus Coding of n-tuples can be obviously defined by iteration of ( , ).

x.

1.2. We inductively put 9( t ) " - t and ( t l , . . . , tk+i} "-- ((ti, . . . , t k ) , tk+i). If 1 _ O-~ #g - k.

PROOF. By fixed point theorem, we find a term h such that h - ) ~ g A z . D O ( g z ) z ( h g ( z + l ) ) , and we choose # ' - ) ~ g . ( h g O ) . Assume that gk -- O A V m < k. gm > O. Then hgk - k and h g m - hg(m + l ), if m < k . By induction we verify h g k - h g O - k. [] 3.5. DEFINITION. A partial number-theoretic function F" w - ~ w is representable in a theory ~T (in the language L) iff there is a closed term f such that: F ( n l , . . . , nk) ~_ n iff ~ F f n l ' - " nk -- n; ( n l , . . . , n k are arbitrary natural numbers; ___ is Kleene's notation and F ( n a , . . . , nk) "~ n means that F ( n l , . . . , nk) is defined and has value n).

3.6. THEOREM. The partial recursive functions are representable in OP. PROOF. S U C , KO, )~Xl...)~Xn.X i represent the initial functions successor, constant-zero, projections (respectively). The recursor and minimalization operators exist by 3.3-3.4; the substitution operator is immediately available by )~-abstraction. F1 3.6.1. REMARK. The representing combinators in 3.6 can always be chosen in normal form and such that, if F ( n l , . . . ,nk) diverges, then f ~ i . . . ~ k - f 2 - ()~x.xx)()~x. xx); cf. w below and narendregt, cit. ,p.179. By 3.6, we denote the standard primitive recursive number-theoretic predicates (e.g. the ordering relation on w) by their customary symbols. Strictly speaking, if P is a primitive recursive predicate, P x stands for the quantifier-free formula f x - O , where f is a term representing the characteristic function of P. It is also clear that OP can provably formalize the standard facts of elementary recursion theory s la Kleene. In the following, we shall adopt the bracket notation { a } ( x ) ~ _ y without distinguishing it from its formal presentation in OP. We conclude with a few observations. 3.7. First of all, the distinction between operations and functions in the set-theoretic sense has interesting conceptual consequences. Let Church's thesis be the statement" CT

Vf(f " N ~ i. ~ qnVm({n}(m) ~ fm)).

1.3]

E l e m e n t a r y Recursion Theory

21

Then CT is consistent with the basic theories we consider in this book, even if full classical logic is used (see 4.11 below). 3.8. It is well-known (see Barendregt 1984, Hindley-Seldin 1986) that numerals, successor, predecessor, definition by cases on numerals and pairing are representable in the theory CL of pure combinatory logic. By CL we here understand the subsystem of O P - , formalized in the sublanguage of s which only contains the function symbols Ap, K, S, the predicate - , variables and logical operators. The only non-logical axioms of CL are COMB and (-~K - S). Here follow the basic steps. (a)

Let T " - g and F " - K I ( I is the identity combinator). Pairing: P A I R x y " - )~u.uxy; L E F T x " - x T ; R I G H T x "- xF. Then CL ~- T x y - x A F x y - y A ( P A I R x l X 2 ) i - x i (i - 1, 2).

(b)

Numerals:

0 " - I; S U C - ~ " - P A I R - ~ K ;

PRED-~ "- LEFT-~;

Z~ "- (RIGHT~)FT.

Then, for arbitrary n, m E w, CL F- Z0 - T A Z ( S U C - ~ ) - F and CL F- -~ 0 - S U C - ~ A ( S U C ~ - S U C - ~ ~ ~ - -~) (apply -~K - S). By fixed point, choose G-

and let D else G ~ -

AxAy.(Zx)(Zy)((Zy)(Zx)(G(LEFTx)(LEFTy)))

A x A y A a A b . ( G x y ) a b . Then CL proves that, if ~ - ~ , G ~ F. Hence by the properties of T, F we are done.

T,

Notice that, once we choose to enlarge combinatory logic by standard numerals 0, S U C O , etc., and we assume S U C as a primitive constant, we are forced to introduce P R E D and D: without them, it would be impossible to define a number-theoretic recursion operator (see Curry et al. 1972, vol.II w 13.A.3, theorem 2). 3.9. On f u l l d e f i n i t i o n by cases. Let DIS be a new constant satisfying the axiom 3.9.1

V x V y ( ( x - y A DISxy - 0) V (-~x - y A DISxy - 1)).

Then 3.9.1 is inconsistent with CL. P R O O F (folklore). Let N e g ( x ) - DISxl. By 2.3 we can find an e such that e - i e g ( e ) . Hence we have that e - T implies e - i e g ( 1 ) - O, and - ~ e - T implies e - Neg(e) - 1. F1 It follows: 3.9.2. CL plus the statement " e v e r y t h i n g is a n u m b e r " is inconsistent.

Introducing Operations

22

[Ch.1

Indeed, if we apply the above trick to D, we get an e such that -~Ne. 3.9.3. There cannot exist an injective operation f from the universe into the natural numbers (define by D an operation h such that h x - 1 , if f x - fO and hx - O, if -,fx - f0; any fixed point of h leads to a contradiction).

g4A. The Church-Rosser t h e o r e m We are going to construct term models for the non-extensional theory OP of operations. The strategy is well-known and it relies upon a fundamental result of Church and Rosser (1936). In order to ensure that the given theory of rules is consistent, we prove that c o m p u t a t i o n s - regardless of the various patterns we may follow- give unambiguous results. This technique regards the equality relation, as inductively generated by an asymmetric reduction relation, which splits the computation process into basic atomic steps. For a thorough treatment of the subject, we send the reader to Barendregt (1984), Hindley-Seldin (1986). In the following we deal with the term fragment of the basic language s which contains individual variables and the individual constants K, S, SUC, P R E D , O, D, P A I R , L E F T , R I G H T . Terms are inductively generated by application from variables and individual constants. (w

4.1. DEFINITION OF REDUCTION. (a) The reduction relation > is the smallest binary relation among terms, which satisfies the clauses below" (i) (ii) (iii) (iv) (v) (vi) (vii)

> is reflexive ( R ) a n d transitive (T); > is preserved by application, namely: t > s, t ' > s' imply t t ' > ss';

K t s >_ t; Strs > ts(rs); L E F T ( P A I R t s ) > t and R I G H T ( P A I R t s ) PRED(SUC-~) > -if; n f i f i t r >_ t; D-~-~tr > r, if n # m;

> s;

(A) (K) (S) (P) (SUC) (n.1) (D.2)

(b) We then read t > r as "t reduces to r" and the clauses (iii)-(vii) are called proper reductions.

(c) The terms which are on the left (right) of the proper reductions are called redexes ( contracta). (d) A term t is normal, or in normal form (in short t E NF), iff no subterm of t (t included) is a redex. For instance, every numeral or basic constant is normal; however, there are terms without normal form, the most

1.4A]

23

Church-Rosser Theorem

typical a m o n g t h e m being gt -()~x.xx)()~x.xx). A reduction s t a t e m e n t t >_ s can be conveniently regarded as a derivable formula of a formal system, where (R) and proper reductions play the role of axioms, while the inference rules are (T) and (A). Hence t>_ s holds iff there exists a derivation in tree-form which is locally correct with respect to the axioms and the rules, and which has t _ s at its root; thus we can recursively assign a length to reductions: 4.2. D E F I N I T I O N (reduction with length), t >_ n s iff (i) 0 - n and t _ s is a proper reduction or an application of (R), or (ii) there are k, rn < u E w and either t >_ k r, r >__m s, for some r, or t' _ k r', t" --> m r" and t - t' t", s - r' r", for some t', t", r', r". (In the following sections of this chapter - is literal identity). 4.3. T H E C H U R C H - R O S S E R P R O P E R T Y CR CR states that the reduction relation _ is directed (confluent)" for all t, t', t" if t > t' and t > t", there exists a term s such that t' > s and t" > s We verify that C R holds for a relation R E D , whose transitive closure is >_. 4.4. (i) t R E D n s is defined by replacing everywhere _ by R E D in 4.2 and by omitting the transitivity rule in 4.2 (ii); (ii) t R E D s " - " t R E D I r s, for some k G w". Obviously, t __ s holds iff t >_ k s holds for some k. 4.5. LEMMA. t ~ s iff there t 1 R E D t 2 , . . . , t k _ 1 R E D t k.

are t l , . . . ,t k such that t l - t ,

t k - s and

The proof is i m m e d i a t e by induction on the length of the derivation. If we replace >_ by R E D in 4.3, we have a statement of the Church-Rosser property for R E D . It is also clear that we can obtain, by simple diagram chasing: 4.6. LEMMA. I f C R holds f o r R E D , C R holds f o r > . 4.7. L E M M A ( A n a l y s i s of R E D )

(i) (ii) (iii)

I f C is an arbitrary constant and C R E D t,

then t -

C.

I f C - K , S , D and C s R E D t , then there is a t' such that s R E D t' and t - C t ' . I f C - S , D , P A I R and C t l t 2 R E D t t i R E D t ~ ( = 1, 2) and t - C t l t ' 2.

, there are t], t'2 such that

Introducing Operations

24 (iv)

[Ch.1

I f C - D and C t l t 2 t 3 R E D t, there are t'l, t'2, t'3 such that tiREDt~(i-1, 2, 3) a n d C t ' i t ' 2 t'3 - t .

P R O O F . (i) C R E D t can hold only by (R). (ii) Let C s R E D O t: since C ~ P R E D , L E F T , R I G H T , we must have applied (R) and we are done. Let C s R E D m t be derived by (A): we have C R E D a t ' a n d s P E D kt'', for some n, k, t', t" such that m > k , n and t - t't". One application of (i) yields the conclusion. (iii) If C t l t 2 R E D o t, t must have the form C t l t 2 , because C ~ K. If m, k < n and C t 1 R E D ks1, t 2 R E D ms2, we can find by (ii) a t~ such that t l R E D t ' 1 and s I - C t ' 1" hence we have t~, t ~ - s 2 , such that t i R E D t~, ( i - 1, 2) and t - Ct'lt'2. (iv): by similar arguments resting on (iii). 13 4.8. T H E O R E M .

CR holds f o r R E D .

P R O O F . Assume t R E D n s l , t R E D ms2: we produce a term r such that s 1 R E D r, s 2 R E D r. We argue by induction on l = n + m . If 1 = 0 and (R) is applied on both sides, choose r - s 1 - s 2. If there is a proper reduction, choose r as the result of the contraction. Let l > 0. Case 1. One of the given derivations has length 0. By symmetry, it is not restrictive to assume t R E D o s 1. We analyse its derivation ~1" 1.1 (R) is the inference applied in ~1: choose r - s 2. 1.2. (R) is not applied. We have to distinguish a few subcases. 1.2.1. Let (K) be the inference applied in ~1" Then t - K t l t 2 and s I = t 1. Since m > 0, the last inference of the derivation ~2 (of t R E D m s2) must be (A): K t 1 R E D k rl, t 2 R E D n r2, where f i r 2 = s 2 and k, n < m. By 4.7 (ii), there is a t~ such that r I - K t ' 1 and t 1 R E D t"l. t'l is the right choice. 1.2.2. Let (P) be the inference applied in ~1" for definiteness, assume that we apply left projection. Then t - L E F T ( P A I R tit2); in ~2 we have, for some rl, r2, L E F T R E D rl, ( P A I R t l t 2 ) R E D r 2.

Hence by 4.7 (i) and (iii), r I - L E F T and we find t~, t~ such that t i R E D t~ ( i - 1, 2 ) a n d r 2 - P A I R t'1 t'2. Choose r - t~. 1.2.3 Let (S) be applied in ~1" Then t - S t l t 2 t 3 and s I - tlt3(t2t3); by 4.7 (iii), and noting that ( A ) i s applied in ~2, we find t~, t~, t~ such that , t i R E D t i, (1 _< i _< 3) and s 2 - S t l, t 2, t 3. Choose r _ tl,t,(t,t,3)" 3~ 2 1.2.4 Let

(D.1)

be the

last

inference

of ~1" then

t R E D t I and

Church-Rosser Theorem

1.4A]

25

I I I t-D-ff-fftlt2, t1 s 1. By 4.7 (iv), we find t3, t4, tl, t 2I such that I I I I I s 2 -- D t 3 t 4 t i t 2 and ~ R E D t'3, ~ R E D t'4, t I R E D t'a, t 2 R E D t 2. Since n is I ~ I I normal, t 3 t 4 -- ~: hence we can choose r - t 1. If D.2 is applied, t - D ~ t l t 2 with ~ distinct from ~ and s I - t 2. By normality of ~, ~ and 4.7(iv), we find a t~ such that t 2 R E D t ' 2 and I s 2 R E D t 2.

1.2.5. Assume t - P R E D ( S U C ~ ) s2-PREDs' , where ( S U C - f f ) R E D s ' : s ' - S U C ~ and we can choose r - ~.

and s 1 - - n . Then for some s', but S U C ~ is normal, hence

Case 2. Assume n, m > 0: hence (A) is the last inference in both derivations. We m a y suppose 2.1 t I R E D k t'x, t2 R E D m

t'l' and t - txt2, s I - t'lt'{, where k, m < n;

2.2 t i R E D p t'2, t 2 R E D q t'2' and s 2 - t~ t~', where p, q < m. But k + p, m + q < n + m and by IH, there are t', t" such that t '1 R E D t', t 2' R E D t ' a n d t 1" R E D

By r -

(A) , t't".

81 - -

t'lt'1' R E D t't"

and

"'"~2~2

t"

, t 2" R E D

s 2 [ t E D t't":

t"

.

hence

we

choose

13

4.8.1. R E M A R K . It is essential to the proof that D be restricted to numerals and, more generally, to closed normal terms (see Church's calculus of A&conversion in Church 1941, Barendregt 1984). There are counterexamples to C R in the case, where P A I R satisfies surjectivity, namely P A I R ( ( L E F T t ) ( R I G H T t ) ) > t ( s e e Barendregt 1984, p.403). However, this last reduction gives rise to a consistent convertibility relation by w If we apply 4.8, 4.7 and the definition of normal form, we get 4.9. C O R O L L A R Y . (i) C R holds f o r > . (ii) I f t ,I t l l E NF and r _>_ t', r _>_ t" then t' and t" coincide ( u n i q u e n e s s of normal form). 4.9.1. R E M A R K . (a) A notion of reduction which enjoys CR, can be defined for the A-formalism, plus ~- and ~-conversion (see w (b) There exist several reduction strategies. However, for combinatory logic, there is a standard reduction procedure SR, whose main virtue is condensed in the Standardization Theorem (Curry-Feys 1958). If t has a normal form, then SR terminates and yields the normal form of t. (c) The property of having a normal form is recursively enumerable, but not recursive, by a classical result of Church.

Introducing Operations

26

[Ch.1

~4B. T e r m Models We are ready operations.

to

introduce

the

syntactical

models

of the

theory

of

4.10. D E F I N I T I O N (i) C T M := { t : t is a closed term}; T M " - {t 9t is an arbitrary term}. (ii) The open term model T M is the structure {TM, 9 , - , where the e m p t y set interprets the truth predicate T and

C,N,T), m

1. 2. .

4.

e - {K, S, D, P A I R , L E F T , R I G H T , SUC, P R E D , 0}; 9 9T M 2 ~ TM is the operation of juxtaposition of terms (i.e. application); = C T M 2 and t - s holds iff t > r and s > r, for some term r; N _C T M and t E N iff t > g, for some n C w.

(iii) The closed term model C T M is the substructure of TM, whose support is TM. 4.11. T H E O R E M . Let ~ " - C T M or T M . Then ~ is a non-trivial model of O P + C T , but ~ is not a model of OPA (for notations 2.6, 3.7). P R O O F (sketch)..At, is non-trivial, because K, S are normal and ~ falsifies K - S (by unicity of normal form). Again the Church-Rosser theorem ensures that - is transitive and that ( N t A t - - s ~ N s ) holds in RAt,. In .At,, -- preserves application, simply because > is closed under (A). The axioms on the special constants and N are true by definition of > and since the numerals of ~ are isomorphic to the standard numbers. Consider the closed term 12K - S ( K 1 ) K where 1 - S ( K ( S K K ) ) . Then -~K - 12K holds in ,A1, because the given terms are different normal terms; thus the axiom MS.2 of OP,~ (see 2.5) fails in ,At,. Assume that f " N--+N (see 3.3) is true in All,; we define F" w--+w by F(n)-m iff f g > ~ . By hypothesis, F is total and its graph is trivially recursively enumerable" hence F is recursive. T h i s - t o g e t h e r with the fact that ,A1, is a model of OP and OP provably formalizes the standard results of elementary recursion t h e o r y - yields the soundness of CT. [3 4.12. C O R O L L A R Y . Let ~ " - C T M (i) (ii)

- ~ I- t - ~

or TM. If t and s are closed terms,

iff O P F t - s ;

~l, l= N t iff OP F t - g ,

for some numeral g.

P R O O F . F r o m right to left, apply 4.11. As to the opposite direction, it suffices to verify that t _ r implies OP F t - r. 13

Term Models

1.4B]

27

The proof of CR is constructive and it can be carried out in the system PA of first-order Peano arithmetic (PA is described in the appendix). Hence we have" 4.13. THEOREM. Arithmetic.

OP

is

interpretable

in

the

system

PA

of Peano

4.13 can be sharpened, once we realize that CR is (at least) provable in PRA, the system of primitive recursive arithmetic. This remark naturally leads to a subsystem of OP, which is tailored for PRA. Let 3(+) be the smallest class of L-formulas which is generated from atoms of the form Nt, t - s by means of A, V and existential quantification; 3(+)-NIND is Ninduction schema restricted to 3(+)-conditions. If we define OP 1 := O P - + 3 ( + ) - N I N D , we can prove: 4.14. THEOREM. If f is a combinator such that OP 1 F f : N ~ N , defines a primitive recursive function.

then f

Of course, the proof depends upon a careful formalization of 4.11 (for more details, see Troelstra 1973, Troelstra and van Dalen 1988, J~iger and Strahm 1994, and the Appendix). 4.13-4.14 can be further strengthened by adding to OP and OP 1 some truths of the closed term model CTM. An important example (to be applied in Ch.VI) is the enumeration axiom: EA

3 f V x 3 y ( N y A f y -- x).

EA holds in CTM, because there is a closed term which enumerates CTM (cf. Appendix). 4.15. THEOREM. Theorem 4.13 (4.14) remains true if we replace OP (OP1) respectively by the system O P + E A + C T (OPI+EA). We do not know whether Church's thesis can be conservatively added to O P I + E A . We also mention that there are consistency results involving continuity of extensional operations, encoding type-2-functionals (see Beeson 1985, Troelstra and van Dalen 1988).

Introducing Operations

28

[Ch.1

w The graph model In this section we describe the classical graph model Pw, due to Plotkin and Scott, its recursive submodel R E and Engeler's DM-models. Pw, D M and R E verify OPA, i.e. OP plus the Meyer-Scott axioms (cf. 2.5) and hence they are models of (the extended) A-calculus with ~-conversion, but without full extensionality (cf. 2.6). The construction of R E can be carried out in OP and this fact yields another method to interpret OP plus the MeyerScott axioms into Peano arithmetic (see w While R E validates Church's thesis, Pw is a model of a strong choice principle AC N on natural numbers" O P + A C N yields a model of full second-order arithmetic. Let us fix a few preliminaries. First of all, we let a, b, c, d, x, y, z, u range over elements of P w - { x ' x C_ w}, where w is the set of natural numbers. Of course, if a, b E Pw, a - b stands for extensional equality; n, m, k, p, i, j range over w. We also adopt the lambda notation informally, i.e. to name (set-theoretically defined) functions. If T is the Kleene predicate, W k - { n " 3 m T ( k , n , m ) } - the k-th r.e. set (r.e -- recursively enumerable) and R E - {x" x E Pw A 3n( W n - x)}. We also put ( n , m ) - l ( n + m ) ( n + m + l ) + m ; $x$y.(x, y) is a primitive recursive bijection of w • w onto w. We also define a canonical enumeration of the finite subsets of w:

e n - { n o , . . . , n p _ i } , provided n 0 < ... < up_ 1 and n -

~

2 hi, and e0 - 0 .

i 1; if n -- 1, we simply omit the index).

Clearly we have in pure logic :

Propfunk+l(f) ~ VxPropfunk(fz ) (k > 1);

8.3.1.

hence we can restrict our attention to unary propositional functions. We now investigate the closure properties of Prop under standard logical operations and the behaviour of T, whenever T is restricted to Prop. Abbreviation: Prop(A):= Prop([A]). 8.4.

LEMMA. (i)

M F - proves:

Prop(A), whenever A = (-~)Nx, (-~)x = y; Tx--. Prop(x);

52

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[Ch.2

(ii) Prop(z)~ Prop(Tz)~ T(Prop(z)); (iii) Prop(z)~ Prop(Prop(z)); (iv) Prop(z)~ Prop(--,z); (v) Prop(z) A Prop(y)---, Prop(x A y) A Prop(z V y); (vi) Prop(z) A (Tz --, Prop(y))---, Prop(z ---,y); (vii) Prop fun(f)---, Prop(V(f)) A Prop(3(f)); (viii) Prop(x,y)~ Prop(z) V Prop(y) (, = V, A,---,); (ix) Prop(Q(f))~3zProp(fz) (Q = 3, V); (x) ,F(Prop(z)). PROOF: straightforward application of 8.2 and T-axioms. As to the final point, if FProp(z)is assumed, we have F T z A F F z (8.2(ii)), whence Fz A Tz (by T.2.1), against consistency T . 6 . 0 8.4.1. REMARK. 8.4 (x)implies T(Vz(Prop(z)---, f z ) ) ~ T(Vf); this means that internal truth (i.e. truth with respect to T) disregards quantification on propositions; thus, there is no hope to produce propositions by means of Vx(Prop(x)---,... ), except for trivial cases. We stress that the internal truth predicate is partial and that 8.4 (viii)(ix) cannot be improved by replacing V, 3 in the right member of the implications with A, V respectively; for instance, there are disjunctive propositions with a member which is not itself a proposition. Therefore the behaviour of logical operators on Prop is non-strict. By 8.4 (x), the notion of proposition is essentially external and positive; we cannot come to know that p is not a proposition by adopting the semantical schema embodied by T. 8.5. PROPOSITION. MF-proves: (i) 3x(-,Prop(x)) A -,Propf un(Iz.[Prop(z)]); (ii) 3z3y(Prop(x V y) A-,(Prop(x) A Prop(y))); (iii) 3f(Prop(3f) A--,VxProp(fz)). PROOF. (i) We consider the fixed point L of )~x.[Fz], i.e. L - [FL] (apply 2.2). Then Prop(L)implies both TL and FL (by 8.2 (i)), against consistency. If .kz.[Prop(x)] were a propositional function, we could conclude by 8.4 (x) Vx.TProp(x), which contradicts the previous result. (ii): choose y - [ 0 - 0] and x - [L].

II.8]

Operationsand Reflective Truth: Simple Consequences

(iii): choose f -

53

~y.[Fy]. Vl

On the other hand, T is consistent and complete on propositions and satisfies the standard Tarski conditions; indeed, the essential content of 8.2 and 8.4 can be summarized as follows: 8.6.THEOREM. MF- proves: (i) Prop(A) A (TA ~ A), whenever A = (-,) x = y, (--)Nx; (ii) P r o p ( x ) ~ Prop(-,x) A (T-,x ~ - , T x ) ; (iii)

Prop(x) A (Tx~Prop(y))---,Prop(x ~ y) A (T(x ~ y ) ~ ( T x ~ Ty));

(iv) Prop fun(f)---, Prop(V f) A (T(Y f ) ~ Y x T ( f x)); (v) Prop(x)--, Prop(Tx) A ( T ( T x ) ~ Tx). 8.6.1. REMARK. (i) 8.6 shows that MF-essentially contains the (classical) theory of Frege structures (see Aczel 1977, 1980). (ii) The Curry paradox (Curry 1942). We cannot consistently add to M F - a strengthened introduction axiom for implication, which omits the hypothesis Prop(x)in 8.6 (iii): ME- + ( , ) i s inconsistent, where ( , ) i s the statement

(Tx ~ Prop(y))--, ((Tx -~ T y ) ~ T(x ---,y)).

(,)

Indeed, we can find c such that c-c---, y ( c - FP(,~x.[x---, y]), see 2.3)and clearly Tc---, Ty (by 8.2 (ii), consistency and ---logic). If we assume (,), we can infer T(c---,y), i.e. Tc, whence Ty by 8.2(iv): contradiction (choose

[0-1]). m

As to the general Tarski schema T A ~ A, it can be justified "from left to right" and also for positive conditions. 8.7. DEFINITION. (i) A formula B is T-free if T does not occur in B. (ii) The collection T-Pos (T-Neg) of T-positive (T-negative) formulas is inductively generated by the following clauses: 1. each e-atom is both T-positive and T-negative; each atom of the form Ts (-~Ts)is T-positive (T-negative); 2. if B is T-positive (T-negative), ~B is T-negative (T-positive); 3. if B, C are T-positive (T-negative), then so is B A C; 4. if B is T-positive (T-negative), then so is VxB.

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[Ch.2

8.8. T H E O R E M (i) (ii)

The soundness schema: M F - proves ( T A ~ A), for arbitrary A; if A is T-positive (T-negative), M F - minus consistency proves: A ~ T A (-~A---, FA, respectively);

(iii)

if A is T-free, M F - minus consistency proves T A Y T-~A.

P R O O F . (i): by induction on A and by considering the form of B whenever A - - - l B . If A is an atom, we apply T.1 and T.2.1, while, if A is a conjunctive or universally quantified formula, we use IH~ T.4.1, T.5.1 plus /?-conversion. If A = - ~ T t , we apply T.2.2 and consistency; in the remaining cases, we make use of T.4.2, T.5.2 coupled with IH. (ii): by simultaneous induction on the definition of T-Pos and T-Neg. (iii): by ( i i ) a n d tertium non datur. F! We conclude with a simple, but useful duality property, whose semantic content will be made clear in Ch.VII (w 8.9. DEFINITION 1. ^ is the (unique) map of the basic language s into itself such that (i) ^ is the identity map on e-atoms and (Tt)^ =--,Ft; (ii) ^ commutes with the logical operations: ( A ^ B ) ^ = A ^ ^ B ^, ( W A ) ^ =

W(A^),

(-,A) ^ =

2. Put COMP := Vx(Tx V T-~x)(Completeness); NMF ( = the neutral MF) is MF minus CONS, where CONS = T.6; MF ^ := NMF + COMP. As usual, N M F - is NMF without N-induction.

3. x=~y := (Tx ---, Ty) A (Fx ~ Fy) and xc~y := ( x ~ y ) A (y=~x). Then CONS ~ (COMP) ^ and COMP ~ (CONS) ^, provably in NMF (use axiom T.3); more generally, we can easily check 9 8.10. LEMMA. NMF proves: (i) A ~ A ; (ii) ( ~ y ) ^ ~ ( y ~ ) ; ( ~ r

(~r

8.11. T H E O R E M (Self-duality of NMF) For every A, NMF F A iff NMF F A ^. The same holds if the restriction axioms RES are added to NMF. P R O O F . By the previous lemma, it is enough to check the theorem from

II.9A]

Type-free Abstraction, Predicates and Classes

55

left to right. The verification runs by induction on the length of the formal proof of A in NMF; the induction step is immediate by definition of ^ and IH. If A is an axiom, either it is self-dual (i.e. equivalent to its ~-transform, like NIND, T.3) or it can be proved by the axiom lying on the same line in the statement 7.10 (e.g. (T.2.1) ^ requires T.2.2). F! 8.12. COROLLARY. For every A, MF F A iff MF^F A ^. MF and MF ^ have

the same T-free theorems and hence they are equiconsistenr NMF is a possible axiomatic counterpart of the four-valued approach to semantics (Belnap 1977, Woodruff 1984, Visser 1984), according to which self-reference leads to underdefined (neither true nor false), as well as to overdefined (both true and false) sentences. For a general account of the NMF-models, we send the reader to Ch.VII.

w9A. Type-free abstraction, predicates and classes We will show that M F - s u p p o r t s a reasonable theory of type-free abstraction. To this aim, we observe that internal truth yields a wellbehaved notion of general predicate application (in short predication), and that the underlying combinatory structure grants a systematic notation for partial predicates defined by abstraction. Furthermore, if we identify total predicates withpropositional functions, we obtain a rich domain, satisfying natural closure conditions for abstraction. Henceforth, we shall adopt Feferman's terminology by using the shorter term class instead of propositional function. Of course, as we already know from the previous section, there exists a stumbling block in any theory of abstraction, based on such an identification" the notion of propositional function (or class) is itself non-total and this is an essential limitation for deriving impredicative fragments of second-order logic. On the other hand, the limitation is not surprising, in view of the reductive, predicativistic interpretation, which is suggested by the C - m i n i m a l model of w7. 9.1. DEFINITION (i) (Xl...Xn)~Ty :- T(YXl...Xn); (ii)

(Xl...xn)-~y - F(YXl...Xn);

{Xl...xn: A} := ~Xl...)~xn[A]("the n-ary predicate defined by A");

(iii) Cl(y):= Yx(xrly V xfiy) ("y is a class"); CL

:=

Note that e l ( y ) = Prop fun(y). We also recall (see 8.10):

xVVy := (Tx ~ Ty) A (Fx ~ Fy) and AC~B := [A]c:~[B].

Extending Operations with Reflective Truth

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[Ch.2

9.1.1. REMARK. (i) The definition does not ensure the injectivity condition

[~] =

[u~v]--,

9 = u ^ y = v

(,)

If (,) is needed, choose P D := ~xy. Ix = x A y = y A yx] and define xrly := T ( P D x y ) , x-~y := T-~(PDxy). Then (,) is met and we can prove in N M F - the formula: T(PDxy) ~ T(yx) A T~(PDxy)~

T~(yx).

(ii) Of course, 77, ~, { } might be accepted as primitive symbols of s and the definitions of 9.1 (i) would become axioms. A similar choice might be advisable in applications, or if one wishes to avoid combinatory logic (see appendix I and Ch.XIV). 9.2. PROPOSITION. (i) The Abstraction principle AP: for every formula

A, N M F - (i.e MF- minus consistency) proves: VUl... Vun((Ul... Un)r]{Xl... Xn 9A} ~ A[x 1 "- Ul,... , Xn "-- an]); (ii) NMF- proves:

((=1... ~ , ) , y ~ T[(~I... ~,),y]) ^ ((~1... ~,)~Y ~ F[(~l... ~,),y]); (iii)

NMF- F- (uT]{x" A} ~ TA[x : - u]) A (u~{x" A} ~ FA[x "- u]);

(iv)

NMF- F- T[Cl(x)] ~-, Cl(x) ~-, x~CL;

(v)

M F - }- Cl(x) ~ (-(y~x)~y-~x).

PROOF. (i) Assume n - 1 " then T(ur]{x" A } ) ~ T ( A x . [ A ] ) u ~ T A [ x "-u] (by T.2.1, fl-conversion and [A][x " - u ] - [A[x "-u]]). A similar argument works for F (we need T.2.2.). (ii)-(v)" left to the reader. [3 By the Russell paradox, there exist predicates, which are not classes, and the notion of class does not determine a class. More generally: 9.3. PROPOSITION. Let r -

{x" ~xrlx }. Then:

MF- ~ ~3~(Ct(~) ^ W(~,~ ~ ~,~)) ^-~3y(Cl(y) ^ W(u,~ ~ Cl(~))). PROOF. Let x be a class such that Vu(urlx ~ urir). Then by AP we have: xrlx ~ x~r ~ x-~x, whence by consistency ~(T(xrlx ) V F(xrlx)), i.e. ~Cl(x). If y is a class, which exactly contains all classes, b - {x" x~y A-~x~x} is a class and br]y: hence br]b~(bqyAb-~b)~b-~b, i.e. b is not a class" contradiction !El n-ary predication can be reduced to unary one by adding parameters: 9.4. LEMMA (Parametrization). NMF-proves:

II.9A]

57

Type-free Abstraction, Predicates and Classes

VXl... VXn+m((Xl... g::}(xn_l_l... X n + r n ) r l { U n + l .

Xn+rn)rl{Ul...

Un_kr n 9 A } r

. . un+rn:A[Ul " - X l , . . . , ttn . -

xn] }.

We now consider some useful approximations to the naive abstraction principle, i.e. to versions of AP where r is replaced by the standard biconditional. 9.5. DEFINITION (i) A formula B is elementary in the list X l , . . . , x n iff B is built up from e-atoms, negated e-atoms, T-atoms of the form trlx i and their negations -~trlxi (1 < i < n ), by means of A, V ,Vy, 3y (y ~ {Xl,...,Xn}); (ii) a formula B is quasi-elementary in x l , . . . , x n iff B is built up from e-atoms and negated e-atoms, arbitrary T-atoms, and negated T-atoms of the form -~t~x i (1 < i _< n), by means of A, V and Vy, 3y (y ~ {Xl,...,Xn} ). 9.5.1. REMARK. (i) B is T-positive iff B is (up to logical equivalence) quasi-elementary in the empty list of variables. (ii) If B is elementary, then B is trivially quasi-elementary; moreover, the negation of an elementary formula is always elementary (up to logical equivalence). The notion of elementary condition for type-free languages is adapted from Feferman (1975). We say that B is (quasi-) elementary tout court iff B is (quasi-) elementary in some list X l , . . . , x n. With the notions of 9.5, we obtain a useful generalization of 8.8 (i)-(ii): 9.6. LEMMA (i) Let A be quasi-elementary in Xl,... , x n. Then: MF(ii)

F CI(Xl) A . . .

A Cl(xn)---+

(A ~ TA).

If A is elementary in Xl,... , xn,

MF- F C l ( x l )

A...

A Cl(xn)--+

Prop(A).

PROOF. (i) By 8.8 and induction on A, using the hypothesis on Xl,...,Xn, whenever A - - , t q x i. (ii): by (i), classical logic and 9.5.1 (ii). F! Lemma 9.6 and the abstraction principle 9.2 immediately imply 9 9.7. COROLLARY. (i) If A(V, X l , . . . , x n ) is quasi-elementary in

Xl,...,Xn,

M F - ~ Cl(xl) A ... A Cl(xn)--+ Vy(y~{v: A(V, Xl,...,Xn) } ~-+A(y, x l , . . . , X n ) ). (ii)

If A(v, Xl, . . . , Xn) is elementary in Xl,... , Xn, M F - F Cl(xl) A . . . A Cl(xn)--+Cl({v: A(V, Xl,...,Xn)}).

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[Ch.2

Corollary 9.7 yields the so-called elementary comprehension schema, in short EC (Feferman 1975). It may be asked whether classes are closed under a strengthened schema, where "elementary" is replaced by some reasonable notion of "second-order condition", e.g. the formula A admits a standard interpretation in second-order logic. In Ch.VII, we shall prove that a second-order impredicative comprehension schema is consistent with MF, but there are models (e. g. the inductive model of w which falsify it. 9.S. DEFINITION (i) Vxrlt.A "- Vx(xrlt --. A);

3xrlt.A "- 3x(xrlt A A);

(ii)

"f is a family of classes indexed by a " : - Vxrla.Cl(fx);

(iii)

{{Xl,...,xn)" A(Xl,...,xn) } ": - { x ' x - ( ( X ) l , . . . , (X)n) A A((x)I , . .., (x)n)} (for (x)i, see w

(iv)

E(a, f) "- {(x, y)" xTla A yrl(fx)} ( - generalized direct sum or join);

(vi)

I I ( a , f ) : - {g" Vxrla. (gx)rl(fx)} (-generalized product).

(iii) is justified in N M F - w i t h pairing axioms: 9.8.1.

(al,...,an)rl{(Xl,...,xn): A(Xl,...,xn) } r A[x 1 := a l , . . . , x n := an].

9.9. PROPOSITION (The Join Principle J). CL is closed under generalized sums over families of classes, indexed by classes. Formally, MF-proves:

(i) w(~,r,(b, f ) ~ 3~3y(~ = (~, ~)^ ~,b ^ y,(f~))); (ii) Yx~lb.Cl(fx) A Cl(b) --, Cl(E(b, f)). PROOF. (i) is an immediate application of 9.7 (i) and 9.5.1. (ii): let f be a family of classes indexed by the class b and let:

A(u) := 3x3y(u = {x, y) A xrlb A yrl(fx)). By T.1, Cl(b), 8.2 (ii)-(iii), T.4.2, 9.2 (v)we get:

FA(u) ~ VxVyF(u = {x, y) A xrlb A yrl(f x)) VxVy(u r (x, y)V F(xrlb) V F(yrl(fx)))

wvy(,., 7: (~, y) v-~(~,Tb) v y~(f~)) wvy(,_, = (~, y) ^ ~,Tb ~ ~,~(f~)) w v y ( u = (~, y) ^ =,Tb ~-~(~,,7(.f~))) ~A(u). Together with (i) and 9.2 (i), this yields:

Operations on Predicates and Classes

II.9B]

59

-~(u~)E(b, f)) ~-~A(u) ~ FA(u) ~ u~E(b, f). [3 9.9.1. REMARK. 9.9 proves the so-called join principle J; 9.9 and 9.7 show that MF contains Feferman's system EM+J for explicit mathematics (Feferman 1975, 1979; Beeson 1985; cf. appendix II). As an exercise, the reader may verify the dual principle for II: 9.10. PROPOSITION (Closure under generalized products). MF-proves: (i) C l ( b ) ~ Vg(g~lII(b, f ) ~

Vx~lb. (gx)~l(fz));

(ii) el(b) A VxTib.Cl(fx) ---, Cl(H(b, f)). Similar arguments prove that Prop, the notion of (internal) proposition, is closed under infinitary conjunctions and disjunctions in the following sense: if f is a family of propositions indexed by any class c (for instance c = {x: Nx}), there exist propositions A { f x : xTIc}, V { f x : x~lc}, satisfying:

T( A { f x : xrlc}) ~-, Vxzlc.T(f x);

T( V { f x : x~lc)).-. 3xrlc.T(f x).

w9B. Operations on predicates and classes

We extend the standard operations of the algebra of (extensional) classes and relations to the general domain of partial properties. In particular, each definable predicate can be generated starting from four primitive predicates by means of eight predicate operations. 9.11. DEFINITION 1. Initial Predicates: ~Pe := {(~,y,z):

OD := { ( ~ , y ) :

9 = yz);

~ = y);

N :- {x:Nx}; ~-:= {x:

Tx}.

2. Basic Operations: Singleton

{a} := {x : x = a} ;

Complement - a

:= {x: -~xTla};

Intersection

a f-1 b "- {x" xqa A xrlb};

Domain

dom(a) := {x: 3y.(x,y)~?a}.

Extending Operations with Reflective Truth

60

[Ch.2

3. Combinatorial operations: Expansion

Exp(a) :-- {(x,y): y~a};

Converse

Cony(a) :- {(x,y) : (y,x)rla};

Cycle

Cyc(a) :-- {(x,y,z) : (z,x,y)ria};

Transpose

Tress(a) :- {(x,y,z): (x,z, ylria};

(x, y, z, a, b denote distinct variables; remind that ( x , y , z ) - ((x, y),z)). 4. We say that CL is closed under a given n-ary operation H, provably in a theory ~l', if C l ( a l ) A . . . A C l ( a n ) ~ Cl(H(al,..., an) ) is provable in ~';

5.

a - b :- Vx(xria r xrib);

6.

a C_ b "- Vx(x~a - , x~b) and a - eb "- a C_ b A b C_ a. {Clearly, if a and b are classes, then a - e b ~ a - b)}.

7. EXPL is the collection of s which is inductively generated by the clauses: ~PP, DD, N, Y E EXPL; if t is a variable or a constant of .5o_, {t} E EXPL; if t E EXPL and s E EXPL, then dora(t), tM s , - t , Conv(tl, Exp(t), Cyc(t), Trans(t) are elements of EXPL. If t E EXPL, we say that t is an explicit predicate of .5. The subcollection ELP of elementary predicates is inductively generated as EXPL, except that we omit Y from the initial clauses of EXPL, and we add the condition" if t is a variable, Exp(t) E ELP. A trivial application of elementary comprehension 9.7 (ii) yields" 9.12. LEMMA. CL contains f~DD, DD, N and is closed under the operations of 9.11.2-9.11.3, provably in MF-. 9.13.THEOREM (Explicit abstraction) (i) For every formula A of s for every n, we can effectively define a term vn(A) E E X P L with F V ( r n ( A ) ) - F V ( A ) - { x l , . . . , x n } , such that, provably in MF-:

vn(A ) - {(xl,...,xn)" A}.

(*)

(ii) Assume that A is elementary in u (where u may be a finite list of variables): then the term rn(A ) can be chosen in ELP. P R O O F . (i) The argument parallels the well-known class theorem for Gbdel-Bernays set theory. We proceed by induction on the built-up of A. The inductive step is easily handled by means of complement, intersection and domain; therefore we only need to find, for each n, a predicate rn(A),

Operations on Predicates and Classes

II.9B]

61

satisfying (~) above, whenever A - Nt, Tt, t I - t 2. On the other hand, t 1 - t 2 is equivalent (mod r to 3 y ( y - t l A y - t 2 ) . Now we can find, uniformly in r, n, i with 1 _< i _< n, a term er'n(r ) such that:

crin(r)- {(Xl,...,Xn)" X i - - r } . Hence:

{(Xl,. ..,

t2} -

Xn > 9tl _

~.n-I- 1 d o m ( ~ nn-t-1 + l ( t l ) fq~n+](t2)).

The definition of ~r/n(r) can be reduced to the construction of elementary predicates I~PPn(i , j , k ) and ODn(i, j), where 1 < i, j < n such that:

f~pPn(i,j,k)-

{<Xl,...,Xn). x i - X j X k }

ODn(i,j ) - { ( x i , . . . , X n ) "

x i -xj}.

(1)

In turn, (1) is verified by induction on n with the help of ~PP, 0I) and the combinatorial functions (details are in appendix II). A similar argument works if A - N t or T t where, of course, we must use N and 3-. (ii) It is enough to deal with the atomic case A(Xl, u) - tou , where Xl, u are distinct variables, possibly free in t. But there is an elementary predicate (r2(t) -- {(Xi,X2>" Z 2 -- t}; if we choose ezt(u,t) - dom(a2(t) f3 E x p ( u ) ) , ext(u, t ) i s elementary and ext(u, t ) - {xl" trlu }. [3 By application of elementary comprehension generalized sums and products, we easily have:

and

closure

under

9.14. PROPOSITION. C L is closed, provably in MF-, under the operations defined in the following list.

1.

Pair"

{a,b} : - {x" x - a V x - b};

2.

PirectSum:

a@b:--{(x,y)'(xT?aAy--O)V(x~bAy--1)};

3.

Cartesian Product:

4.

Exponenlialion: [a--,b] "- { f " Vxqa.(fx)qb};

5.

Universe and empty class:

m

v.-

6.

m

a | b "- {(x, y)" xrla A yrlb};

-

0

.

-

Generalized union and intersection:

n fz .- {u. Vz,Tb.u,7(fz)}, U far {u 93xrlb. u,l(fx)} , xob xrlb provided b is a class and f : b ~ CL. 9-

9.14.1. REMARK. In analogy with the generalized closure condition of P r o p under implication, it holds, provably in MF-:

Extending Operations with Reflective Truth

62

[Ch.2

Cl(a) A (3x(xrla) ~ Cl(b))----, Cl([a---, b]) A V f(frl[a ~ b] ~ Vxqa. (fx)rlb). We end up the section by considering a natural question: is there any reasonable counterpart at the level of predication of the set-theoretic power set operation? Let us try two obvious alternatives:

P(a) "- {x" x C_ a}; P+(a) "- {x" xrlCL A x C_ a}. Unfortunately, the former is unmanageable, while CL is not closed under the operation P+: 9.15. PROPOSITION (i)

M F - F ~VyVx(x C_ y ~ x~?P(y));

(ii)

M F - F Vz(z~?P+(y)~ Cl(x) A z C_ y);

(iii)

MF-F~Vx(CI(x)~CI(P+(x)).

PROOF.(i): choose x - CL and apply 9.3. (ii) is obvious. As to (iii), choose y - V ( - universal class) and verify that Cl(P+(V)) implies Cl(CL): contradiction !O Clearly, both operations ~x.P(x) and ~x.P+(x) are C_-monotone (e.g. a C_ b implies P(a)C_ P(b), etc.); however, in the present framework we can also define an intensional power set operation" 9.16. DEFINITION (Weak power set). P w ( b ) " - {x " 3 y ( x - y

gl b)}.

Then by elementary comprehension 9.7 we have, provably in MF-: 9.17.

Cl(Pw(b)) A Vu(u C_ b---, 3v(wlPw(b) A v -- e u)) A

where - e is the extensional equality of 9.11.6. The interest of Pw(b) is limited by the fact that, even if b is a class, we cannot predict whether any given subproperty of b is itself total. A weakening of the power set operation along constructivistic patterns, looks more promising and it leads to consider the decidable subclasses of a given class x. If c is a class, a decidable subclass of c has the form {x" x~c A f x -- 0}, for some f" c ~ 2 (here 2 - {0,1}). 9.18. DEFINITION. P d ( a ) " - {y" 3f3y(y - {z" xrla A f x -- 0} A f~[a --, 2]}. Then we have: 9.19. PROPOSITION. MF-proves:

Cl(c) ~ (Cl(Pd(c)) A Vb(brlPd(c ) ---, Cl(b) A b C_ c)).

Fixed Point Theorem for Predicates

II.10A]

63

We underline that Pd(c), for certain c's, can be very large: it is consistent to assume that P d ( N ) i s a model of second-order comprehension (cf. 5.13).

w10A. The fixed point theorem for predicates We know from the previous sections that the logical universe described by MF-, though not well behaved under impredicative quantification over classes and propositions, is closed under natural infinitary constructive operations, which go beyond the limits of elementary logic. We now consider the problem of inductive definitions; we shall see that M F - can prove the existence of solutions to a number of recursive conditions, but in general such solutions cannot be shown to define total predicates, or to be extremal (i.e. minimal or maximal). The main tool is the property-theoretic analogue of the second recursion theorem. Indeed, if we combine the fixed point theorem for operations and the abstraction schema, we get the simple and fundamental: 10.1. T H E O R E M (Fixed point for predicates) (i) Let A(x,y,v) be a formula with the free variables shown only. We

can f i n d - uniformly in A - a is v, such that M F - p r o v e s :

term IxyA(x,y,v),

whose only free variable

IxyA(x, y, v) = {x: A(x, IxyA(x, y, v), v)} A A Vu(u~IxyA(x, y, v) r A(u, IxyA(x, y, v), v)). (ii) /f A(x,y,v) is quasi-elementary in v (a fortiori if A is T-positive, see 8.7), and we assume that v is a class, r can be replaced by ~-~. PROOF. Choose:

IxyA(x,y,v) := FP(Ay.{x: A(x,y,v)}). Hence u q l x y A ( x , y , v ) c ~ A ( u , IxyA(x, y, v), v) by 2.3 and 9.2. The second part follows with 9.7. D 10.1.1. NOTATION. 10.1 introduces a new variable binding operator I. If x, y are clear from the context, we shorten IxyA(x,y, v) to I(A, v); we also let I ( A , v ) : = I(A) if the dependence on the variable v is not explicitly needed. Of course, the definition of I makes sense in the general case where v is a finite (possibly empty) list of variables, apart from the bound x and y. 10.2. Examples (i)

The notion I S of "ileralive set" (or "hereditary class").

64

Extending Operations with Reflective Truth

Let I T S ( x , y ) := Cl(x) A x C_ y and choose I S ( x ) : = xrII(ITS). have, by 10.1(ii) and exploiting the classhood of x: 10.2.1

MF- ~ I S ( x ) ~

[Ch.2 Then we

Cl(x) A Vz(z,lx ---, IS(z)).

In chapter V we shall prove that I S (actually, a generalization) yields a well-behaved set-theoretic universe. (it) The intensional notion of (finite) type over N. Let T P ( x , y ) be the formula:

[(= = N) v 3c3b(~ = [b-~ ~] ^ b~y ^ c ~ ) v ~b3c(~ = b | c ^ b ~ ^ c~y)]. If T Y P E ] " - I ( T P ) , we get, provably in MF-, that T Y P E / c o n t a i n s N and is closed under Cartesian product and exponentiation. (iii) Let:

The construclive second number class O.

O~d(., y ) . -

=~N ^ [(. - T) v 3 k ( . - 2 k ^ k~y) v

V 3kVn3m({k}(n) ~_ m A mTly A x -- 3k)] (for the notations 2 k, 3 k, recall the convention 3.6.1). If O "- I(Ord), M F proves the fixed point axiom 10.2.2.

Vm(m~O ~ Ord(m, 0)).

By inspecting the examples above, we are naturally led to lift to the present framework the notion of (positive) operator, which is familiar from the standard theory of inductive definitions (Moschovakis 1974). 10.3. DEFINITION (i) A formula A(v) is operative in v iff either v does not occur in A or A belongs to the least class of formulas, which is inductively generated by means of A, V, Vy, 3y (y distinct from v) from formulas of the form Nt, -~Nt, t - s, -~t - s, tTlv, Tr, -~Tr, provided v does not occur in t, r, s. (it) If A is a formula, which is operative in v and T-positive, the term ~ v . { x l . . . x n : A ( x l , . . . , X n , V)} is called an operator, n being the arity of the operator; A may contain free variables ~ {Xl,... ,Xn, v}. (iii) An operator Av.{xl...Xn: A(Xl,...,Xn, V)} is elementary if every T-atom, which occurs in A, has the form t,lv. (iv) An operator ~ v . { x l , . . . , Z n : A(Xl,...,Xn, V)} is existential iff no universal quantifier occurs in A, except possibly for universal bounded number quantifiers of the form Vm < n (see 3.1).

II.10A]

Fixed Point Theorem for Predicates

65

10.3.1. CONVENTION. We henceforth identify any given operator with the formula defining it; thus we simply call A(Xl,...,Xn, V) an operator in v tout court, and X l , . . . , x n show its arity. In the following we only consider operators of arity 1 (or 2 at most; this is not restrictive by pairing). The idea is that any given operator defines a monotone operation (with respect to C_ of 9.11), transforming relations (represented by) v into relations (represented by) { X l . . . x n ' A ( X l , . . . , X n , V)}. The notion of existential operator is suggested by the formula involved in the definition of T Y P E I , while Ord(x, y) is elementary and I T S ( x , y) is not. In set theory (or in a suitable fragment of second-order logic), the Knaster-Tarski theorem grants the existence of fixed points; here we have, by induction on the build-up of the given operator and by 10.1 (ii)" 10.4. PROPOSITION. ( i ) / f A(v) is operative in v, then in pure logic

A(v) A v C_ u ~ A(u). Hence if A(x, v) is an operator, )~v.{x" A(x, v)} is C_-monotone, provably in MF-: a C b ~ {x" A(x,a)} C_ {x" A(x,b)}. (ii) For every operator A(x, v), I ( A ) : - IxvA(x, v) is a fixed point, provably in MF-:

Vu(urlI(A ) ~ A(u, I(A)). 10.4.1. REMARK. We warn the reader that the class of T-positive formulas, which are operative in the variable v, is not closed under substitution" e.g. A(x, v) - XrlV is an operator, but A(v, v) - VrlV is not. Proposition 10.4 raises two natural problems. We first wonder whether there exists a more mathematical characterization of operators, which does not refer to the syntax: the question will be answdred in the positive in Ch. IV, where we shall prove that operators coincide with extensional (or C_-monotone) operations and have a natural topological interpretation. A second question concerns the apparent weakness of 10.4(ii): it only offers implicit solutions to the given condition and states nothing about the minimality (or the maximality) of the solution I(A), which is essential to argue by (generalized) induction on the given I(A). In the next chapter, we shall see that such limitation is essential: MF- is consistent with different hypotheses on I(A) and hence MF- is formally unable to distinguish among them. However, proposition 10.4(ii) already establishes a simple link with standard subsystems of second-order arithmetic ( - analysis, in short).

Extending Operations with Reflective Truth

66

[Ch.2

Indeed, let us consider the standard language s of PA ( - first-order arithmetic), expanded by a new unary predicate symbol P; an arithmetical operator is simply a formula A(x,P) of s which is positive in P (i.e. A is logically equivalent to a formula, built up from atoms of the form Pt, t = s, --t = s by means of V, A, 3, V), and has the free variable shown only. s , the language of the elementary theory of inductive definitions, is obtained from the language of PA, by adding a distinct unary predicate symbol I A, for each arithmetical operator A(x,P). A

10.5. DEFINITION. ID 1 is the first-order theory in the language Z(ID1) , which contains Peano arithmetic PA (with the induction schema, extended to all formulas of s ) and the fixed point axiom:

FP(A)

VX(IAx~A(X, IA)),

for each arithmetical operator A (of course, A(X, IA)results from A(x,P) by replacing each subformula of the form Pt with IAt ). It is clear that every arithmetical operator becomes an elementary operator in the sense of 10.3 above, once we replace "Px" with "xr/v" (we can assume that v is a fresh variable); thus, if we choose N as the range of individual variables and we interpret "IAX" by "xrlI(A)" , we readily obtain: A

10.6. PROPOSITION. ID 1 is interpretable in MF. Since I131 has the same arithmetical theorems as the subsystem EI-AC or, equivalently, Predicative Analysis of levels < e0 (cf. Aczel 1977, Feferman 1982 and Ch.VIII), we also have a lower bound on the arithmetical content of MF (which turns out to be sharp by the proof theory of chapters IX-XI). At this stage of the investigation, it becomes essential to calibrate the strength of the number-theoretic induction available; so we explicitly introduce two finitely axiomatized subsystems of MF, which restrict NIND to arbitrary properties and to classes respectively. 10.7. DEFINITION induction).

(Subsystems of MF with restricted number-theoretic

(i) Set Clos(y) "- (Orly) A Vx(xrly~(x+l)77y). Then" Property N-induction P-NIND: Class N-induction CL-NIND: (for C see 9.11.6).

Vy(Clos(y)~N C_ y); Vy(Clos(y) A C l ( y ) ~ N C_ y);

II.10A]

Fixed Point Theorem for Predicates

(ii) MFp "- MF-+P-NIND;

67

MFc := MF-+CL-NIND.

Later, we shall prove that the inclusions MF c C_ MFp C_ MF are proper; while MF c is proof-theoretically equivalent to OP, MFp already proves the consistency of OP. If we restrict our attention to existential operators, then we can give a standard inductive definition of minimal fixed points. 10.8. LEMMA. Let A(x,u) be an existential operator. Then we can find a term Ax.IXA such that, provably in MF" (i) Vn(I~ -- { x : - , x -

x} A InA+1 -- {x" A(x,I~4)});

(ii) VnVp(n < p---, InA C I~); (iii) Vx(A(X, I A ) ~ 3kA(x, IkA)). PROOF. (i)-(ii): we apply recursion on N (see 3.2), N-induction and proposition 10.4(i). As to (iii), proceed by outer induction on the build up of A; here it is necessary that A is existential and we apply the true arithmetical schema:

Vn(n < m--~ 3 k B ) ~ 3jVn(n < m ~ 3k(k < j A B)), which is provable by induction on m. O Now choose:

10.9. THEOREM. MF proves (for B arbitrary, A existential operator): (i) gx(A(x, IA) ~ x~7IA); (ii) Vx(A(x,B)---, B(x))---~ Vx(xOI A - , B(x)). PROOF. (i): by 10.8 (iii), (i). (ii)" if we assume the premise of (ii), VnVx(xrlI~4---, B(x)) is derivable by means of N-induction and monotonicity. O 10.9.1. REMARK. If the existential operator defined by A(x,u) is elementary in u, then by property N-induction, elementary comprehension 9.6 and 9.14, we have: MFp F VnCl(InA)A CI(IA).

(1)

Under the same hypothesis on A, (1)implies that 10.8 (ii)-(iii) are provable in MFp. Hence, 10.9 (i) together with

Extending Operations with Reflective Truth

68

[Ch.2

is already derivable in MFp. (1) also holds if every T - a t o m occurring in A ( x , u ) - e x c e p t those of the form trlu-is of the form sr/r where r can be proven to be a class. Hence 10.9(i) and the special case of 10.9(ii) with B(x) = xTlu are derivable in MF p "

wlOB. Applications to semantics and recursion theory The fixed point theorem 10.1 turns out to be a significant tool in general: in most applications, one only needs the existence of a solution and not its m i n i m a l i t y / m a x i m a l i t y . We illustrate the theme by showing the existence of a partial satisfaction predicate for 2, and then by proving two abstract versions of well-known results, due to Rice and Myhill. First of all, OP is obviously sufficient (see w to carry out a primitive recursive arithmetization of the L-syntax; so, we can fix an effective GSdel numbering [ ] and we let [E] stand for the (canonical term of L representing the) GSdel number (in short gn) of the expression E. For later applications, it is also convenient to define the satisfaction predicate over GSdel numbers of arbitrary terms, possibly encoding formulas via [... ]. To this aim, we say that t is a formula-term if t = [A], for some formula of s 10.10. D E F I N I T I O N (i)

If f is any term, f( + ) is the operation defined by x

f ( xi ) i - x

,

f ( xi ) n - fn, if - - , n - i .

Here we suppose that n, k, i range over N. Clearly f(/=)is well-defined, uniformly in i, x (apply definition by cases on N ; see 2.1, NAT.2). (ii) By 3.6, we may assume that there are formulas and terms in the T-free part Lop of the language s defining the following notions:

Ter(x) := x is the gn of a term; F o r ( x ) "- x is the gn of a formula of s Fort(x): = x is the gn of a formula-term [A]; f t r ( z ) := the gn of [A] if z is the gn of A (i.e. ftr([A]) = [[A]]); vr(x) := the gn of the z-th variable; tr(z) := the gn of [Tt], if z is the gn of t; id(x, y):= the gn of [t = s] if z = It], y = Is]; nat(z) := the gn of [Nt] if z = [t]; neg(z) := the gn of [-~A], if z is the gn of [A]; and(x, y):= the gn of [A A B] (x gn of [A], y gn of [B]); all(x, y):= the gn of [VziA ] (x = [zi] , y gn of [A]);

Applications to Semantics and Recursion Theory

II.IOB]

69

app(x, y):-- the gn of (ts) (x gn of t, y gn of s).

10.11.LEMMA. There exists a term Val such that, provably in OP-, Yal([tl, f ) -

t[x o "-- f O , . . . ,x n "-- f-i],

for every term t with free variables in the list Z o , . . . , x n. In particular, O P - proves

Val([[A]], y ) -

m

[A[~ o " - f 0 , . . . , ~ n " - f n ] ] ,

for every L-formula A with free variables in the list Zo,... , z n.

PROOF: we define an operation Val such that" Yal ([vi], f) - f~; Val ([c], f ) - c if c is a constant; Val ([t~l, f ) -- Val(rtl,/)Va/(r~l, f). Val is well-defined by means of the operation D and fixed point theorem 2.3 (it is essential that the operation F 1 is N-valued and that the language has a finite number of constants). E!

In order to introduce the satisfaction predicate, let S ( z , v ) formula saying that z has the form (m, f) and 1. 2. 3. 4. 5. 6.

m -mmm m m -

be the

id(n, k) and Val(n, f ) - Val(k, f ), or nat(n) and N ( Y a l ( n , f)), or tr(n) and T ( Y a l ( n , f)), or nag(n) and -~( 0, whence [c I > 0, which implies O(a?b)l=-~Tc (by 14.2 (ii)), i.e. O(all,)l= Fc with (.). Assume the right hand side of (1). Then clearly b E O(dtl~) and b must be in p-form by R E S . If b is in E-form, [b[ - 0 and trivially b < c. If

94

Inductive Models and Definability Theory

[Ch.3

c is in E-form and O(~)l-Fc, o r c is not in p-form, then c ~ O(Jll~) by consistency of O(~1~): hence b __ c (by 14.2.2). Ad (2)" similar argument, using the assumption from right to left. [3 14.3. T H E O R E M (Uniform Ordinal Comparison). There exists an operator G(u, v) in lhe language 2., such thai if.At, i= OP-, then for every b, c E M:

b w, it follows, by inspection of the definition of the term models in 4.10:

232

[Ch.8

Levels o f Truth

39.3. LEMMA. The sets CTM, N "-- {t" t E CTM, t >-~, for some ~} (cf. 4.1 for > , ~ numeral), the application function , " CTM • the interpretations of the basic constants O, S U C , P R E D , P A I R , R I G H T , D of OP are all elements of La, for every c~ > w.

LEFT,

In order to interpret the local truth predicates of TLR, a delicate step is the choice of a denotation for the function symbol LT; we must find an injection I N of t into CTM, which does not spoil the self-referential abilities of the T i's and is "reasonably" definable. Once I N is available, the model building only requires the well-known closure of admissible sets under El-recursion and El-inductive definitions (see Barwise 1975, pp. 24, 124, 208), plus the fact that L is an admissible ordinal, which is limit of smaller admissibles. 39.4. LEMMA (i) The predicate A d ( ~ ) " -

"c~ is admissible" is uniformly AI(L/3 ) for [3

limit > a).

(ii)

The operation fl ~ fl+ -

the least admissible > fl

is uniformly AI(La) , for c~ limit of admissibles.

(iii)

Let r o - ~ and rc~ - least admissible 7 > r~3, for every fl < c~, whenever c~ > O. Then the ordinal sequence (ra : c~ < i~) is uniformly

/kl(Lrs). (iv)

t is the least c~ such that v a - c~. In particular the restriction of v to L is ~l(Le).

PROOF. (i): by standard techniques of formal set-theoretic semantics and the well-known uniform Al-definability of the operation 5~--~L~ (see Barwise, cit.; Devlin 1977). (ii): apply (i). (iii): by (i)-(ii) and closure of admissible sets under ~l-recursion. (iv): easy consequence of (iii). 1"1 39.4.1. REMARK. If w < a < t

and a is admissible, a - f l

+, for some

We can now state the main fact needed for interpreting LT: each admissible fl with w < fl _ ~, such that IN[rc~" r a ~ w is total and injective, for every 0 < a ~ 2 iff PA F-Cons(~ , where PA is Peano arithmetic (cf. appendix to Ch.I) and Cons(OJ") is a standard formalization of the metamathematical statement "~ is consistent" in the arithmetical language. ~ 1 and ~ 2 are proof-theoretically equivalent (in short, ~ 1 = ~2) iff zJ"1 > ~2 and ~ 2 > ~ ~ 1 > r iff 03"1 > ~ 2 and not ~1 -- ~2" If ~ 1 > ~ we say that ~1 is proof-theoretically stronger than ~ 2. We now state without proof the following known results: 40.4. T H E O R E M (i)

A1-CA 0 - II~-CA 0 and II~-CA 0 > ATR 0.

(ii) ATRo=_Predicative Analysis (in the sense Feferman 1964, cf.Ch. XI). Moreover ATR o > ~E1-DCo.

of Sch~ttte

1977,

(iii) ~ - D C o > A~-CA o. (iv) A1-CAo - ACA o - PA. (v) RCA o - PRA. (ii)-(iv) above can be obtained as a corollary of the proof-theoretical analysis of Ch. XI (but see Feferman-Sieg 1981, for (i), (iii)-(iv), Friedman, Simpson and Mc Aloon 1981, for (i) and Simpson(199?)for 40.4(v)). It is to be mentioned that each instance of A]-CA is derivable in ATRo; however E]~-DC is unprovable in ATR 0 (actually ATR 0 + ~ - D C is strictly stronger than ATRo, by theorems of Friedman). We are now ready to state the promised interpretation result. We recall that PWp is obtained from the system PW c of 16.1 by replacing numbertheoretic induction for classes with number-theoretic induction for properties.

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[Ch.8

40.5. T H E O R E M (i) A T R 0 is interpretable in TLR. (ii) (iii)

~ - D C 0 is interpretable in PWp + EA. A]-CA 0 is interpretable in P W c.

PROOF. The proof is easy, as the essential work was already carried out in earlier sections. (i) We first define a translation * of Z2 into the level-free part of the language of TLR. Informally speaking, we simply verify that N, plus the subclasses of N, is a model of ATR in TLR. More formally, we choose combinators 0, -, +, ~, in order to interpret the basic function symbols of s (we adopt the same notation). Hence we can inductively assign to each s t a term t* in the language s ( - the operational fragment of s with the same free variables. Moreover, if t - s, tcX, t < s are atoms of s we put ( t - s ) * - ( t * - s * ) ; (tcX)*-(t*rlx) (x fresh variable); (t < s)* = (t* < s*) (the second occurrence of < being a canonically chosen 2.op-definition of < ; see 3.6). We then extend * to arbitrary formulas of Z2 by stipulating that * commutes with 9, A and (VXA)*

-

Vx(CIN(X ) -+ A*), (VxA)* - Vx(Nx--. A * ) - VnA*,

where ClN(x ) := Cl(x)A Vu(u~lx--~ Nx). It is clear that * is a well-defined translation of 2.2 into 2.. Let A be an s with free variables in the list X = X o , . . . , X n , y = YO,'",Yk: then we check by induction on the definition of ATR0-provability: if ATR 0 F A(y,X), then TLR F Ny A ClN(X ) -. A*(y,x).

(1)

It suffices to see that the *-translations of Ax-IND, HI-CA and A T R are provable in TLR. Now (Ax-IND)*and (H1-CA) * become instances of class N-induction and elementary comprehension and hence are provable in T L R by theorem 37.9. Note also that, if (CIN(X) A WO( < X))* is assumed, then < x encodes a subclass of N which is a pwo. Hence if z is any subclass of N and A ( u , x , Y , Z ) is arithmetical, urlNAA*(u,x,y,z)is elementary extensional in y, z (y, z fresh variables). Now the hypothesis of 38.6.2 are trivially met, and there exists a subclass of N satisfying the *-translation of the ATR-consequent. (ii): apply the elementary dependent choice schema EDC of 20.10 (ii). (iii): we simply apply A-comprehension 16.7 of Ch. IV to the translation of hyperarithmetical comprehension. V1 To conclude, it is time to reconsider the opening problem of the section, concerning the theoretical relevance of TLR and its ability to represent significant parts of mathematical knowledge. The answer is implicit in the

VIII.40]

Levels of Truth and Second Order Arithmetic

243

interpretation result above. Here we freely rely on results of Feferman, Takeuti, Friedman and Simpson (op. cit.). First of all, significant parts of ordinary mathematics, like elementary calculus and countable algebra, can be already developed in conservative extensions of Peano Arithmetic (Takeuti, Feferman) and actually in fragments of arithmetical analysis ACA0, which are not proof-theoretically stronger than primitive recursive arithmetic (hence afortiori in fragments of MFc). A typical example thereof is the Cauchy-Peano theorem CP, asserting the existence of solutions for ordinary differential equations; CP is indeed equivalent to KSnig's lemma for binary trees WKL (modulo RCAo) by a theorem of Simpson (1984). Furthermore, the very principle of arithmetical comprehension is equivalent in RCA 0 to the statement that every Cauchy sequence of reals converges to a limit in R and also to the existence of maximal ideals for countable abelian rings or even to the KSnig lemma for finitary trees. On the other hand, ATR 0 has a good theory of countable ordinals and it proves non-trivial classical results of descriptive set theory. In particular, as Friedman and Simpson observed, ATR 0 is mathematically much more effective then the subsystems of hyperarithmetical analysis and even predicative analysis in the sense of Feferman-Schfitte: there are important consequences of ATR0, which are false in the model of hyperarithmetical sets and hence independent of Predicative Analysis. Here is a sample of significant results. ATR is equivalent (modulo RCA0) to: (i) comparability of well-orderings, i.e. countable well-orderings are comparable; (ii) the Lusin-Sierpinski theorem: every analytic set in the Baire space ww of unary functions w---+w is either countable or has a perfect subset); (iii) the Gale-Stewart theorem: every open game C ww is determined; (iv) the Ulm structure theorem for countable reduced abelian p-groups (Friedman, Simpson and Smith 1984). At the same time, the strength estimate of 40.4 (ii) assigns precise limits to ATR 0 and hence, by the equivalence theorem of Ch. XI, to the theory TLR of truth with levels. For instance, there is no way to prove in ATR o that every arithmetical set of Dedekind reals has a least upper bound, nor ATR 0 proves the classical Cantor-Bendixson theorem (every closed subset of the Baire space is the union of a countable sets of reals plus a perfect set); by contrast these two theorems are derivable in II~-CA 0 (and hence in the extension of TLR of w

Levels of Truth

244

[Ch.8

~41. Consistency of a reducibility principle for classes We wish to have a closer look to the recursion-theoretic model C~, in order to investigate quantification on classes. In the usual inductive models, like O(CTM) (see w C L - { x " Cl(x)) is generally not closed under quantification of classes: the best we can afford is a sort of A-comprehension (see corollary 16.7). However, we shall verify that, if an elementary

predicate of classes, possibly depending on additional class parameters, is non-empty, then the same predicate is already satisfied by some class of any level ~- i (e.g. a solution is to be found in CLi+I, if i has a successor level i + l ) , provided i is an upper bound on the level of the given class parameters. Hence, at least for elementary predicates, quantification on

arbitrary classes is reducible to quantification on classes of a fixed level. But we know that quantification on classes of fixed levels does not push outside the realm of classes. The result we hint at above, is in essence a consequence of the effective nature of C T M and generalizes to the present framework the classical Kleene basis theorem (Kleene 1959). Formally, we consider a reducibility schema for classes RPC: 41.1.

i -~ k A Cli(x ) A 3y(Cl(y) A A(u, x, y))---, 3y(Clk(y ) A A(u, x, y)),

for every Z-formula A(u,x,y) elementary extensional in x, y.

with the free variables shown, which is

The rest of the section is devoted to convince the reader that RPC holds in Ct. The proof combines the classical analysis of 1-!l-sets by means of recursive trees with a straightforward transfer argument from the standard model of Z 2 to C~, coupled with representability of inductive predicates in our language. 41.2. First, we recall the relevant recursion-theoretic notions and results. As usual, n, m , k range over natural numbers; e, f, g denote indexes of partial recursive functions. Par abus de langage, we keep using ( . . . ) for a fixed number-theoretic primitive recursive function, which injectively maps finite sequences of natural numbers into numbers; on the same par, (n)i will denote the corresponding primitive recursive projection, such that, if n encodes (n0,... , nk) and i < k, then ( n ) i - n i. lh(s) is the primitive recursive function which computes the length of the sequence code s. Seq(x) is the predicate "to be a (number which encodes a) finite number sequence", while ( ) " - 0 is the code of the empty sequence. If s, r, range over elements of Seq, the concatenation of r with s is r , s "- (r0,... ,rn,s0, ...,sin) , where r - (%, . . . , r n ) , s - (So,...,Sm). t is a subsequence of s (in symbols t C_ s) iff s - t , r , for some r in Seq; if r # ( ) , we write t C s. If F ' ~ - - - , ~ , we inductively define F ( 0 ) - ( ) and F ( n + l ) - F(n),(F(n)). The expression

A Reducibility Principle For Classes

VIII.41]

245

{e}P(n) " m means that the e-th partial function, which is recursive in the set P C w, converges on n with value m; clearly the Kleene bracket relation can be defined in the standard model by a E~ which is also denoted by { e } X ( m ) ~ _ n and contains a second-order parameter X; TotX(e) := Vn3m({e}X(n)~_ m) means that the function with index e is total on natural numbers, grs is the index for the partial recursive function defined by {g[s}X(s ') := {g}X (s.s') (s, s' in Seq). If s - (n), we simply write gin instead of g[(n). In order to define recursive trees, we consider the arithmetical formula 41.2.1

W ( e , X , Y ) :- TotX(e) A (Vs(-~{e}X(s) ~_ O) VVn((eVn)cY)) ).

Note that W is positive in Y, while X possibly occurs in negated atoms of the form -~tcX. Then we can define:

coW(X) "- VY(Clos(W, X, Y) --+ ecY),

41.2.2.

where Clos(W, X, Y) "- Ve(W(e, X, Y) ~ ecY). By standard arguments: 41.3. LEMMA

W.1

VXW(W( , X, ~(~)1= v x ( v e ( w ( e , x , B)---+ B(e))---+ Ve(ecW(X)---+ B(e))),

w.2

B(x) is an arbitrary s and W ( e , X , B ) results from W ( e , X , Y) by replacing each atom of the form toY by B[x := t].

where

If P is a fixed set C w, ~g(P) denotes the II]-set defined by the formula

W ( X ) in z)(w), when X is assigned P as value. In general, to any index e of a total recursive function and any P C w, we can associate the tree

W'c

0},

which is closed under the subsequence relation C . ~ ( P ) well-founded trees which are recursive in P:

encodes the set of

41.4. LEMMA. If P C_ w, then e E ~ ( P )

to the converse of C,

iff T P is well-founded with respect i.e. there is no function F : w---~w such that for

Let w w - {F" F is a unary function w---+w}; then II]-predicates enjoy a simple, but essential property: 41.5. LEMMA (Normal form). If R C_ w is II] in a given P C_ w, then we

can find a primitive recursive predicate S R such that, characteristic function of P, then"

if F p is the

ieveIs of Truth

246

[Ch.8

R - {n E w" (VG E ww)(3m E w)( w ; thus the definition by cases used in ~1 does not lead out of the class of the Al-operations. It is straightforward to check that, mutatis mutandis, all conditions of 39.15 hold for ~1; hence we can define Cu - ( C T M , ~ I ) and extend the interpretation of 39.16. Then TMA holds as immediate application of II2-reflection of p over the class of recursively inaccessible ordinals. D On the axiomatic side, we propose a few easy consequences of TMA: 9. PROPOSITION. T L - + TMA proves: (i)

Vi3k(i -< k A Inac(k)) (choose f - g - )~x.x in TMA);

(ii)

V x 3 i T i ( f x ) ~ V j 3 k ( j -< k A Inac(k) A V x T k ( f x));

(iii)

f " C L ~ CL. ~ 3 k ( A d m ( k ) A f " CL k ~ c n k ) .

10. PROBLEMS 1) Study the relation with recursion in the superjump functional sJ (see Hinman 1978 for the relevant definition); for instance, is every set X C w recursive in sJ, definable by a class in Ct,? Are the sets C w recursively enumerable in sJ (which are known to coincide with IIl-sets), definable by closed terms of Cu? 2) Charachterize the least ordinal for which the ~r-construction is no more possible. There exist similar ordinals already below the first stable ordinal cr0 (see Barwise 1975, Hinman 1978). For (r0 the lemma 39.5 fails badly: a0 is projectible into w, but there is an c~ < a0 which is not projectible into w. 3) W h a t is the proof theoretic strength of T L - + T M A ? Rathjen (1991) is probably relevant here.

The work of

CHAPTER 9

LEVELS OF TRUTH AND PREDICATIVE WELL-ORDERINGS w w w w

On well-orderings Ramified hierarchies Predicative well-orderings I Predicative well-orderings II

In the previous chapter we announced that the system TLR of reflective truth with levels is equivalent, as to its arithmetical content, to FefermanSchiitte predicative analysis (henceforth FS). In order to prove such a claim, we are going to develop a proof-theoretic analysis of TLR. The first step is to describe the standard primitive recursive well-ordering of type F0, the Feferman-Schfitte ordinal, within the context of TLR. Indeed, we shall work in a fragment MFR(p) of TLR, which includes: 1) the ground system MF c of Ch. II with number-theoretic induction for classes; 2) axioms stating the existence of the ramified hierarchy, generated by conditions of a fixed logical complexity p, along suitable explicitly presented pseudo-well-orderings (in the sense of 38.4). In w we discuss two different notions in MF: pseudo-well-orderings (orderings, which are well-founded with respect to classes, in short pwos) and quasi-well-orderings (orderings which are well-founded with respect to properties, in short qwos). We shall prove that MFp derives the analogue of the Weyl principle for qwos and related transfinite recursion schemata. In w44 we construct a formalized version of the second-order ramified hierarchy %, the classical model-theoretic counterpart of predicativity. w167 contain an elementary presentation of the so-called predicative standard well-ordering of type F 0 and a well-ordering proof within the fragment MFR(p) of TLR. More precisely, we verify that for each c~ < Fo, the segment of type c~ of the standard well-ordering is a pwo, provably in MFR(p); in the special case where a = e0 (respectively Cw0, the first ~critical ordinal) MFR(p) can be replaced by MF c (MFp). The results are optimal by the upper bound theorems of Ch. XI; there, we will establish a constructive consistency proof of the theory TLR (MFc, MFp) within Peano arithmetic extended by transfinite induction up to F 0 (Co, Cw0). For the applicative-minded reader, we mention that, since a few years,

258

Levels of Truth and Predicative Well-Orderings

[Ch. 9

F0, as well as more powerful proof-theoretic ordinals, have found non-trivial applications in the study of term rewrite systems.

w 43. On well-orderings It is well-known that the notion of well-ordering (and more generally of well-foundedness) is essentially second-order and it depends on the extension of the universe 91 of second-order objects, be they sets, predicates or functions over the ground level. In the familiar arithmetical case, this dependence shows up in the non-absoluteness of the well-ordering notion, with respect to the standard predicative interpretation (that is, suitable segments of the hyperarithmetical hierarchy); there are straightforward examples of primitive recursive linear orderings on w, which are HYP-wellfounded, i.e. well-founded with respect to hyperarithmetical sets of numbers, and yet not truly well-founded (see Rogers 1967, Harrison 1968). On the other hand, the whole power set of w is not necessary to test wellfoundedness. By the Kleene Basis Theorem of 41.7, if a linear ordering -< w of w is well-founded within any collection 91 of sets C_ w, containing -<w and closed under the hyperjump operation X ~ ' W ' ( X ) (for ~dY(X), see 41.2.2), then "<w is a well-ordering "in the real world" (compare with the bar induction corollary of 41.13). {Such q.l.'s are properly included by Friedman 1969 in the well-known collection of Mostowski's/3-models, the wmodels of second-order arithmetic Z2, which are absolute with respect to II~-conditions}. It is then natural to see how far we can p r o c e e d - w i t h i n the theories of reflective t r u t h - in dealing with countable well-orderings. As we mentioned in w38, there are at least two possible versions of the well-ordering notion; and we already know that pseudo-well-orderings are pleasantly closed under forms of transfinite recursion, provably in TLR. Below we argue informally; nevertheless, it should be always clear how to work out the results in the indicated axiomatic systems. 43.1. D E F I N I T I O N (we repeat 38.4). (i) Let us first remind that if w defines a binary relation, i.e. w is a property of ordered pairs, we keep using the infix notation x "~w Y in place of (x,y)~lw. Also, Field( -~w) is the term { x : 3 z ( X - ~ w Z V Z - ~ w x ) } representing the field of "~w and LO(-~ w) means that "~w is a linear ordering. If B(x) is a formula with the free variable shown,

Progr( -~ w,B) "- (VxrlField( ~ w))(Vy ~ w x.B(y) ~ B(x)),

On Well-Orderings

IX.43]

259

where Progr( -.4 w,B) is to be read "B is progressive" (relative to -4 w)" If B ( x ) - xrlb, we write Progr( -4 w,b). As usual, T I ( ~ w,B) "- Progr( -4 w , B ) ~ Vx(xrlField( -4 w ) ~ B(x)), while T I ( -< w,b) stands for Progr( -4 w, b ) ~ Field( -4 w)C_ b. (ii) We recall that -<w is called a pseudo-well-ordering-in symbols P W O ( -~ w), and, in short, "<w is a p w o - iff Vb(Cl(b)~ T I ( ~ w, b)). (iii) A linear ordering "<w is called a quasi-well-ordering-in symbols QWO( -4 w), and, in short, ~ w is a q w o - i f f VbTI( -4 w,b)). (iv) A qwo (pwo) -4 w is acceptable iff "~w is a class; -4 w is unbounded (on its field) iff Vx(x~IField( -4 w ) ~ 3y(yTIField( -4 ~) A x -4 w Y))" Given an acceptable unbounded pwo or qwo, we introduce the standard notions of "zero" ( -4 w-least element), "successor" and "limit"" 0.4 w " - the -4 w-least element of Field ( -4 w);

S .4 w(X)"- the -4 w-least element of {y" yT1Field ( -4 w) A x -4 w Y}; Lim(x) iff "x is nor 0 -~w neither a successor". (v)

A q w o (pwo) -4wiS locally decidable via f and h, if h" a ~ a

and

f ' a ~ {0, 1, 2}, (where a - Field ( -4 w)), and for every x in a, we have u

fx -

0 ifx - 0.~w 1 if x is a -4 w-SCCessor; 2 ifxisalimit;

(,)

hx - the predecessor of x, whenever f x -

1.

(**)

A pwo (qwo) is locally decidable iff it is locally decidable via some f and h. If a pwo (qwo) -4 w is acceptable, its field and every initial segment of -4 w (of the form {x: x -4 w Y}) are classes. 43.2.REMARK. The expression "quasi-well-ordering" is used by Crossley (1969) for linear well-orderings of w, which are well-founded with respect to recursive sets. We now show that a strong transfinite acceptable locally decidable qwos. 43.3. D E F I N I T I O N (cf. 9.14, for |

recursion

principle

below).

f " CL ~ CL "- Vx(Cl(x) ---+C l ( f x)); g. Field( -4 w) | CL | CL--, CL ":-- VxVuVv(x~lField( -.4 w) A Cl(u) A Cl(v)---+ Cl(gxuv));

holds for

260

Levels of Truth and Predicative Well-Orderings .-

[Ch. 9

{u.

43.4. THEOREM (Special transfinite recursion along qwos). We can f i n d provably in M F - - a closed term ~h)~fi~w)~z)~x.Re[h,f, -4 w]ZX such that, if h" CL---~CL, f " C L ~ C L and -4w is an acceptable qwo, which is locally decidable via gl, g2, then for every x in Field( -4 w), m

gl x - 0 ---+ Rc[h, f, -4 w]ZX - hz;

(1)

D

gl x -- 1 ~ Re[h, f, -4 w]ZX - f(Rc[h, f, -4 w]z(g2x)); gl x - 2 ---, Re[h, f, -4 w]ZX - {u" 3v(v -4 w x A u~(Re[h, f , -4 w]ZV)); E(Field( -4 w),)~xRc[h,f, -4 w]ZX) is a class, whenever z is a class.

(2)

If hz C C L and f x C_ CL, whenever z, x are classes, then Rc[h, f , -4 w]ZX C_ CL. (3) {NB: for simplicity, we omit the uniform dependence of Rc from gl, g2}"

PROOF. (1). Observe that the operation gl makes possible to define by cases over the field of -4 w" Then the fixed point for operations implies the existence of Rc[h, f -4 w] satisfying the given equations. (2) It is enough to check that, if z is a class, then Vx(xrlField( -4 w) ---*Cl(H(x, z))),

(,)

where H(x, z) - Rc[h, f , -4 w]ZX; then we can apply the join principle of 9.9. But (.) is straightforward by -4w-induction applied to {x" C l ( H ( x , z ) ) } with the assumption on h, f and -4 w" (3) If G(z) - {x" Rc[h, f , -4 w]ZX C CL}, (.) implies xrlG(z) ~ Rc[h, f , -4 w]zx C_ CL.

(**)

Hence we can proceed by -4 w-induction to show Vx(xrlField( -4 w ) ~ xrlG(z)), under the additional assumption of (3). V1 We also mention that a version of the Weyl principle w holds for qwos. With the notations of 38.4, T R ( y , A , -4 w,z) is a shortening for the formula VuVxVu(xrlField( -4 w) ---*(urly(x) ~ A(u, x, yrx, z))). Then we have, by an easy adaptation of 38.6: 43.5. THEOREM (WP for qwos). Let A ( u , x , y , z ) be a formula, which is elementary extensional in y, z with the free variables shown. Then we prove in MF-: (i) if -4w is an acceptable qwo, z is a class, then there exists a class y such that T R ( y , A, -4 w, z) holds.

Ramified Hierarchies

IX.44]

261

(ii) Under the same hypothesis, if y, y' are two classes satisfying T R ( - , A, -.< w,Z), then for every x in Field( -< w), we have

Vu(u,y(.) In general,

the

u,y'(.)).

hypothesis of 43.5 (i) above cannot

be weakened to

PWO( ~ w), unless we substantially enrich MF-, e.g. to TL-. Indeed, by a theorem of Spector and Gandy, there is an elementary condition A such that "<w is a well-founded recursive linear ordering of w iff a solution to T R ( - , A , ~ w ) exists in HYP ( - t h e collection of hyperarithmetical sets C_ w). Thus no HYP-solution (in our case, no solution in CL) to T R ( - , A, -< w) exists in general, whenever "<w is only HYP-well-founded; furthermore, there are recursive pwos on w such that no solution at all exists for T R ( - , A , - < w) (Friedman 1976). In positive form: postulating

closure of CL under elementary transfinite recursion along arbitrary pwos is to require that CL is really much richer than in the simple inductive models of MF-.

w44. IL~mified hierarchies In this section we apply the special transfinite recursion theorem 43.4 to prove the existence of the second-order ramified hierarchy % of classes of natural numbers along any given acceptable unbounded locally decidable qwo. The result can be extended to pwos in the formal setting of TL. Let us first recall an informal definition of %. By L2 we understand the language of second-order arithmetic, introduced in the previous chapter (40.1). Thus L2-formulas are also closed under quantifications VX and 3Y on sets of numbers, and atoms have the form t E X and t = s (t, s terms built up from variables by means of + , 9 and successor function symbols). 44.1. DEFINITION. Let b~ be a family of subsets of w ( w - t h e natural numbers): (i) if

A

is

a

Xl,...,Xk, X1,...,Xn,

formula

of

s

with

free

variables

in

the

set of lists

~f~A[nl,...,nk, P1,...,Pn] stands for the usual satisfaction relation: it is understood that n l , . . . , n k E w are assigned in the given order to X l , . . . , x k and number quantifiers range over w; P1, " " , P n E ~f are assigned to X 1 , . . . , X k and set quantifiers range over the family ~.

262

[Ch. 9

Levels of Truth and Predicative Well-Orderings

P C_ w is ~ - d e f i n a b l e (with set parameters) iff P - {n E w" (w, ~f)I=A[n, Q I , ' " , Qn]}, for some s A(x, X1,...,Xn) Q1,'-', Q , E :f of set parameters. (i) n e f ( ~ f ) -

and

a

(possibly

empty)

list

{ P C ~" P is ~f-definable}.

(ii) We define by recursion on countable ordinals an operation ~o such that % 0 - nef({w}); (where A is a limit),

zJ~a+ 1 --

Def

(%a) and

%~-

U %~3 f~ < ~

44.1.1. REMARK. (i) Def({w})extensionally coincides with the family of the subsets C_ w, which are definable by s with no set quantifier and no set parameter. If P E %0, P is called a r i t h m e t i c a l . (ii) Once one has accepted w and a segment A of ordinals, the definition of % is predicative in the traditional sense: set quantifiers range over countable collections, which are already built up. (iii) We mention a few basic facts about % (for definitions and proofs, see Apt-Marek 1974, Moschovakis 1974, Boyd-Hensel-Putnam 1969, Kleene 1959, Jockusch-Simpson 1975). 1.

% is a hierarchy, i.e. c~ < fl implies %a C_ %~.

2.

There exists a countable ordinal fl0 such that % ~ o -

%f~o+1 and %f~o

is the smallest fl-model of second-order arithmetic Z 2 (Gandy-Putnam). 3.

flo > w~k ( -

4.

Every set of %f~o is A~ and so is %~o"

5.

%

ck -- H Y P

the first non recursive ordinal) and flo is Al-definable;

- the collection of hyperarithmetical sets C_ w.

w1

44.2. DEFINITION.

Recall that a - e b " - Vx(xTla ~ x~lb); if we put

E m b e d ( g , ~1, ~2) "- Va(ar]~l --+ (ga~l~f2 A a - e ga));

then we define: (i)

3'1 _C + if2 "- 3 g E m b e d ( g , ~fl, ~f2);

(ii)

~fl - + ~f2 : - 3'1 _C + ~f2 A ~2 C_ + :fl"

Any pair g, h such that E m b e d ( g , 3'1, 3'2) and E m b e d ( h , :f2, ~fl) is said to w i t n e s s 5~ +:f2" Of course, C_ + is reflexive and transitive and, by definition of E m b e d with - ~ , it follows that, if g and h witness ~ f l - + :f2,

Ramified Hierarchies

IX.44]

263

g is 1-1 onto with respect to - e" Indeed, we have VaVb(ga - e gb---, a - e b); furthermore

Embed(g,~~

~~ ]k Embed(h, ~f2, ~fl )

---. Va(a~f 1---, h(ga) - e a) A Vb(br/~f2 ---. g(hb) - e b). 44.2.1. FACT. M F - proves: C l ( l l ) A C/(~f2) A ~1 C C L A ~2 C_ C L . ~ T(~f2 C_ + 5'1) V F(~ 1 C e-t- ~2)"

( The same holds with

- + in place of

C_ +).

Of course, the relations - +, C_ + are natural strengthenings of extensional equality and inclusion; we need them to capture the extensional features of the ramified hierarchy in our non-extensional framework. We also remind that MFp " - M F - + P-IND, and that property induction P-IND has the form: Clos(a) ---. N C a; here C l o s ( a ) ' - Orla A Vx(x~?a---, (x + 1)r/a); N stands for the class {x" g x } of MF-. 44.3. THEOREM (MFp) We can f i n d - u n i f o r m l y in any given acceptable unbounded qwo "~ w, locally decidable via gl, g 2 - a n operalion Ax.~-POx such that, if x is in the field of -~ w, then (i)

g~-

0 ~ ~ -

D~f({N)) (i.~. iS 9 i~ th~ -~ w-Sight ~ l ~ , ~ t ) ;

gl x -- 1 ~ %x -- D e f ( % g 2 x ) U %g2x (if x is the ~ w-SUCCessor of g2~); g~

-

2~ ~

- (~- 3v(~ -~ ~ ~ ^ ~ ~ ) } ,

(if ~ i~ ~ -~ ,~-limit);

(ii) %= is a class of classes and %x C_ + %u' whenever x ~ w Y" PROOF. The argument requires a number of separate steps and definitions; the essential point is to introduce the operation D e f in our language. First of all, we associate to each primitive function symbol f of s a closed )~-term f*, which formally represents it (see Ch. I, 3.6). We then define a satisfaction predicate S A T [ ~ , f l , f2 ] for s uniformly in any fixed class ~f of classes C N, and in any pair of operations

/1 " N ~ N , f2" N--~ ~'.

To this aim, we fix a G6del numbering G D of s in such a way that all the syntactical notions (term, formula, occurrence of a free variable, etc.) define classes (provably in MFc); G O ( E ) ambiguously denotes the G6del number of the expression E and the term, formally representing it. For2(x )

Levels of Truth and Predicative Wel/-Orderings

264

is the predicate "x is the Ghdel number of an s predicate "x is the G6del number of an s Put

var2(i ) "-- GD(Xi);

ins(GD(t);

[Ch. 9

Ter(x) is the

var2(i)) "- GD(t E Xi);

Iden(GD(t), GD(s)) "- GD(t - s). If f" N---+~ ( ~ . - N or ~f) and b is in it;, then f(~)" N ~ !/; is the operation defined by: f(~b)n- fn, i f i ~ - n , f ( ~ ) i - b ( here f ( ~ ) i s well-defined with definition by cases on N). Moreover val(u, f l ) i s (the formal presentation of) the operation, which associates to the term encoded by u its numerical value under the assignment fl" 44.3.1. Let W ( ~ , f l , f2, b,k) stand for (the formalization of) the following condition (as to logical complexity, cf. Introduction, 5.5)" (i) every element of b has the form (GD(A),fl, f2) , for some L2-formula A of logical complexity < k (k being a natural number), where f~" N ~ N , f 2 " N ~ ; (ii)

(GO(A), f l, f 2)~lb iff either: 1.1. A - (t - s) and val(GD(t),fl ) - val(GD(s),fl); or 1.2. A -

t E X i and val(GD(t),fl)~I(f2i); or

2.

A - -,B and not (GO(B), f l , f2)rl b; or

3.

A - B 1 A B 2 and (GD(Bi) , fi, f2)~ b, (GD(B2), f l , f2)rl b; or

4.

A - VxiB and VkrlN. (GD(B), fi(~), f 2)rlb; or

5.

A - 'r

and VP~I:f. (GD(B), f l,

i b. f2(p))

Then we prove: 44.3.2. There is a term F -

F(~f, f l , f2) such that

(i) MFp F (Cl(~) A ~ C CL A f l " N ---. N A f2" N ~ ~)

--+Vn~IN. (Cl(Fn) A W(~f , f l, f 2, Fn, n)); (ii) for each n E w, ^

c CL ^ f l "

N ^ f 2"

-+ (Cl(F~) A W(~f, f l , f2, F~, ~)). Verification of 44.3.2 (i). Let Co(fl, f2) , C l ( f l , f2) be the terms (in the given order)"

{x" 3u3v(x -- (iden(u, v), f l, f 2) A Ter(u) A Ter(v) A val(u, f l) -- val(v, f l)));

IX.44]

265

Ramified Hierarchies

{z " 3u3i(N(i) A Ter(u) A x -- C~1. As to the uniqueness, assume a = r 2 =r and C a l a 2 > a2, al, r > ill' f12" By P.4, we must have a 1 = fll and hence f12 = a2" [:! 45.5. C O R O L L A R Y (Extended normal form for ordinals < Fo). For every ordinal c~ E (0, Fo), there exist unique Oil, i l l , ' " , O~n, fin, n >_ 1, such that (i)

r

i > ~i, c~i (i E [1, n]);

(ii)

r

1 ~ . . . ~ c/)O~nfln;

(iii)

c~ = r

fll -F... + r

P R O O F : apply 45.4 to the Cantor normal form. E! Below, we temporarily adopt the symbol = means that t and s coincide).

also for literal identity (t = s

Levels of Truth and Predicative Well-Orderings

272

[Ch. 9

45.6. A system of ordinal notations. We consider a term-language which contains: (i) a new constant e. (ii) a concatenation operation 9 and a binary function symbol f . The set T E R of terms is inductively generated from c by closing off under the clause: if tl, S l , . . . , tn, s~ (n >_ 1) are terms, so is f ( t l , sl) 9 . . . , f ( t ~ , s ~ ) . The degree of t E T E R gr(c)

-

is recursively assigned:

0;

gr(:f(tl, S 1 )) - - gr(t 1) + gr(s 1) + 1 and gr(t 1 , . . . , tn) - gr(t 1 ) + ... A- gr(tn) (n > 1). We recursively define the character cr(t) E T E R , for each t E T E R : t C if n > 1; cr(c) - c; if t - f ( t l , s l ) , . . . , f(t,~,sn), let cr(t) ( t I i f n - - 1. We

now

introduce

:< C _ O N x O N .

a

structure C o V - ( C N , :_ f ( t n , Sn) and cr(si) :(. ti, for i -

(ii)

1,...,n.

t ~: s holds iff s ~ c and one of the following cases holds: 1. t - c , or

2. 2.1. 2.2.

3. 3.1. 3.2. 3.3.

t - t 1 , ... , tn, s - s 1 , . . . , s i n , t, s E C N , n - F r o > 2 and n<mandt i-si,foralliE[1,n],or there is a j _< n, m, such that tj :(. sj and t i - si, for all l _ < i < j , or t - f ( t l , S 1), S - f(t2, 82) , t, s E C N and t I - t 2 and s 1 ~: s2, or t 2 ~: t 1 and t ~: s2, or t l :< t 2 a n d s l ~ : s .

Clearly 45.6.1 can be reduced to a s t a n d a r d course-of-value recursion, and it is s t r a i g h t f o r w a r d to check" 45.7. L E M M A . C N , ~. are primitive recursive. T h e n we have:

IX.45]

Predicative Well-Orderings I

273

45.8. LEMMA. (i) C N is linearly ordered by : 0) (wc~+ 1. ~ < r where U - U(a, ~, X) -- r

__., Vfl(iu(/3) ~ iU(r

'

5(X).

46.2.5. LEMMA (provably in MFc). Let p >_10. If X C_N is a class and

~

) is good (see

46.1), then {•" A ~ ( , ) )

is a strongly progressive

class. PROOF. By hypothesis {c~" AX(c~)} is a class. We need a simple fact of ordinal arithmetic (provable in OP; see Schiitte 1977, p.93), which explains the choice of the ordinal terms: ift,<w a.5,~_ 10), we get Ax(0) as a consequence of 46.2.3 (iii) and the fact that ~ Pj y + l ( X ) is good. Next, we show: X(c~ + 1) (X C hi class). Fix 5 > 0 and assume

(2)

Predicative Well-Orderings II

IX.46]

k-

281

(3) (4) (5)

6o c~+2. ~ < w-,/+l.

zX(a);

IW(fl), If c is in ~

where W -

%~(X).

and is progressive, we want

(,)

1)fl).urlc. crI%P(X),

(Vu < r

+

By (1), choose 6 ' > 0 such that where V - - w a + l ' ~ ' < k. Obviously %P(X)C_ CL N is a class, being a segment of %~(X), and it is closed under the map * (u being a limit). If U - % P ( X ) i n 46.2.4, we have with (4): sU(a) is strongly progressive and hence progressive. But sU(a) is in ~ ( X ) because u < k, v is a limit and logical complexity < 10 < p; hence by (5),

IU(r

(6)

+ 1)fl)

has

(7)

w, < ft.

By progressiveness, flrlSU(a), i.e. IU(r + 1)fl), which implies (,), as c is progressive and in U. As to the limit case, we may assume, for X C N, X class: Vn. AX(A[n])(A limit); W-

aJ~oPA+l. 6(X) and t r

(8)

r "k+l" ~ < r "y+I (where 5 > 0);

(9)

IW(fl).

(10)

As in the successor case, let c be a progressive class of W, i.e. c is in U - %P(X), where u - ,))'.6' < n, for some 6 ' > 0. In order to apply the second part of 46.2.4, we show:

VnVfl(#(fl) Indeed, fix n arbitrary, assume Then

a~l%~(X), for

IU(fl)

(11) and pick a progressive class a of U.

some ~ < v and by (1) we can find a 6n such that ~n < b', i.e. ar]CJ}oPA[n]4.1.r

< c~

- Un

Hence, since U - %~P(X)_D Un, we have, by downward persistence, But (8) and (9) imply"

IUn(fl)---+ IUn(r IUn(r Vu < CA[n]fl. urla; this umon over the Un's. " Therefore

holds and completes But (11) progressive. Since u < n, Lu(A)is

for every ft.

IUn(fl). (12)

by progressiveness of a, we conclude that the verification of (11), because V is the and 46.2.4 yield that L v (A) is strongly an element of W (see (9)), whence by (10)

Levels of Truth and Predicative Well-Orderings

282

[Ch. 9

Vu < fl. urlLU(A).

(13)

But Lu(A)is progressive, hence flrlLU(A), i.e. Iu(r since c is progressive. Vi

which yields (.)

46.2.6. First part of theorem 46.2: MFp ~ PWO(a), for each a < tw0. MFp F- Cl(X) A X C_N ~ Good(% m(X)), for each m C a~. P R O O F . Let a < tw0, and choose m such that a < t m 0 . Since wm+2 is a qwo (46.2.3 (v)), %Pm+2(X)is good by 44.3 (X is assumed to be in CLN). x 1(m); for 6 - 1 , we get Vfl(IU(/3) ---. IU(r Then 46.2.5 implies Am+

,

where U - %Pm+I(X). If we choose f l - 0 and we remark that X belongs 0~ to U, we get TI(r

X); but X is an arbitrary class, whence the

conclusion follows by 46.2.1. V1

Proof of the theorem 46.2 (conclusion). By 46.2.3 and 46.2.6, it remains to check

the

case of i f - M F R ( p )

with

p > 10. By the axiom

schema

M F R ( p ) F (PWO(a) A X~?CL A X C_N)--,Good(%P(X)).

(1)

RAM(p), we have for arbitrary a < F 0" We prove (i)-(ii) below by metamathematical induction on n, where a o - w O~n+ 1 - - r

(i)

M F R ( p ) F VX(X~?CL N --, Good(%Pn(X)));

(ii)

M F R ( p ) F PWO(an).

If n - 0, (ii) simply reduces to class N-induction, while (i) follows from (1). Let n - m + 1. By IH we may assume PWO(am) , whence PWO(a m + 2), which implies by 4 6 . 2 . 3 - p r o v a b l y in MF c - PWO(wam+2). Hence, if X is any subclass of N and ~ -

wC~m+ 2, we have by (1) that ~o~(X)is good. But

we can apply lemma 46.2.5 and we get that A X + l ( a ) is progressive, and ITS

hence with PWO(wam+2), AaXm+l(am) , i.e. PWO(r A final application of (1) yields (i) for n - m

) -PWO(am+X).

+ 1.0

46.3. REMARK. F 0 and the applications. H.Friedman showed that the wellfoundedness of the standard well-ordering ~: for F 0 follows from a theorem of Kruskal about well-quasi-orderings (see Gallier 1991); hence Kruskal's theorem is unprovable in Predicative Analysis. On the other hand,

IX.46]

Predicative Well-Orderings II

283

Kruskal's theorem is a powerful tool for investigating term rewrite systems used in computer science. Thus Friedman's result suggests that there may be connections between (segments of) the standard well-ordering ~: of type F 0 and the term orderings involved in termination proofs of term rewrite systems. Indeed, Dershowitz and Okada established interesting relations with proof-theoretic ordinals; for instance, it can be shown that the ordertype of the so-called multiset path ordering on the terms of an alphabet whose precedence ordering is w, is exactly Cw0. For a survey on term rewrite systems, the reader can consult Dershowitz and Jouannaud (1990). The relevance of F 0 for combinatorics and computer science is discussed by Gallier (1991), where the results of Dershowitz and Okada are also reviewed.

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C H A P T E R 10

REDUCING REFLECTIVE TRUTH WITH LEVELS TO FINITELY ITERATED REFLECTIVE TRUTH w w w w

w w

A sequent calculus STLR for a theory of reflective truth with levels Basic properties of STLR Elimination of the full level induction schema Elimination of unbounded level quantifiers The infinitary sequent calculus I T ~ of n-iterated reflective truth Embedding STLR n into I T ~

In semantic form, the main theorem we are going to establish sounds as follows: the first recursively inaccessible ordinal can be replaced by w in the construction of the recursion-theoretic model of w39, insofar as we deal with TLR-consequences of the form

Vi3jVz(Cli(z)---, 3y(Clj(y)A A(z,y))) (A elementary extensional in x, y). Indeed, something stronger will be true: as a consequence of proof-theoretic analysis, we shall prove that the theory T L R of reflective truth, with variable levels and full transfinite induction schema on level ordering, can be constructively reduced to a family {ITS" n E w} of theories of arbitrary finitely iterated truth predicates. In each system I T S , level variables and quantifiers are explained away in favour of a sequence {Tk: k 0 and S T L R ~ F ~n + l

F, then S T L R ~176 F ~n

F"

To avoid repetitions, proofs will be given for the infinitary systems of the next chapter. 49.9. L E M M A (i) If A is an arbitrary s STLR~176 b 0~o--,Progr(-.4 , A) , ViA(i);

{Progr( -~ , A) abbreviates Vi(Vj(j -.< i ~ A ( j ) ) ~ A(i))}. (ii)

S T L R ~176 F- 1< ~' F, provided F is a sequent of the following form:

{-~i ~_k,-,T~t, Tkt}; {i5i};

{-~i~j,-~j_k,i_~k);

{--,i ~ j , - , j ~ i, i -

{3k(iqkAj-
0), F is provably recursive in OF iff there is an B(+)-formula A ( X l , . . . , X m , Y ) of OF (see Ch. I, 4.13), such that: F ( n l , . . . ,nm) -- p implies OP t- A ( ~ I , . . . ,rim,p); OP F- VXl ... V x m ( N x 1 A . . . A N x m --+ 3!y(Ny A A(Xl, . . . , Xm, y)))}. Now 49.18 can be formalized in OP + T I ( w k ) , for each given k > 0 and derivations of length <w k. Therefore the partial recursive function F of the asymmetric interpretation theorem should be at most wk-recursive. These remarks are useful for the conservation theorems of Ch. XI. We conclude with a second problem, concerning a requirement of greater uniformity on F: is it possible to define a majorizing function F, which satisfies the conditions of the asymmetric interpretation theorem 49.18 and only depends on the height a of ~ and m ?

X.50]

A Calculus for n-lterated Reflective Truth

303

w50. The infinitary sequent calculus IT~~ of n-iterated reflective truth

We devise a new system I T S , in which level variables, bounded level quantifiers and level atoms are omitted. 50.1. The syntax of IT~n 50.1.1. The language s of I T ~ is s without" ( i ) b o u n d level variables and bounded level quantifiers; (ii) the function symbol i T ; (iii) the predicate symbols - / , _ . The L-terms of Ln are the level constants of value _ n; the individual terms of s coincide with Z-terms (they are generated without the clause: if i is an L-term, L T ( i ) i s a term). The atoms of s have the following form" N t , t - s, -~Nt, . " t - s (e-atoms); T i t , Fit , ."Tit , ."Fit (T-atoms; t, s are s i is a level constant of s s -f~ are inductively generated from s by means of A, V, Vx, 3x. 50.1.2. Since L T is omitted, we must redefine the map A ~ [A], for we stick to the previous definition (see 36.3), except that [ T i t ] " - ( 7 , ( i , t ) ) , where the boldface occurrence of i stands for constant of s while the overlined occurrence denotes the ( - closed term of .5; see Ch. I), whose value is the value of i.

A Es we let a level numeral

CONVENTION. Unless it is unclear from the context, we keep using the same symbols i, k, j, n, m for level constants and their values. 50.1.3. Simultaneous inductive definition of P O S n and N E G n. (i) P O S n is the least class of s which contains every A of the form t - s, . " t - s, N t , ."Nt, Fit , ."Fit , T i t , -.Tit , for i < n, T n t , F n t and is closed under A, V, V, S; if A E P O S n, A is said n-positive. (ii) N E G n is the least class of s which contains every A of the form t - s , -.t-s, N t , -.Nt, Fit , -.Fit , T i t , -,Tit , for i < n , ."Tnt , -.Fnt and is closed under A, V, V, 3; if A E N E G n, A is said n-negative. (iii) 50.1.4. (i) (ii) (iii)

A is n-separated iff A E P O S n or A E N E G n. DEFINITION of K n ( A ) ( = n-complexity of A E Ln)" K n ( A ) = 0 if A is n-separated; else: K n ( B o C) = m a x ( K n ( B ) , K ~ ( C ) ) + I ( o = A, Y ); K n ( Q x B ) = K n ( B ) + I (Q = V, 3).

For a proper statement of the axioms, it is convenient to recall the obvious finitary generalizations of A and V" 50.1.5.

~ { A i" i m such that I T ~ F Vx(Clm(x ) ---,3y(Cln(y ) A Amn(x,y))),

(1)

where Amn(Z,y) results from A(x,y) by replacing each occurrence of t~lz (t~ly) with t~lmx (t~lnY). By 51.8 and (1)we have

C~l=Vi3jVz(Cl~(z) ~ 3y(Clj(y) A A~j(z, y))).

(2)

By assumption on A, we also have, for m, n E w" ^

cl.(y) ^ Amn(Z , y)---, A(z, y)).

(3)

(2)-(3) yield the required conclusion. 0 However, theorem 51.9 is still unsatisfactory, because it does not make any deep use of the constructive information, associated to the proof tree of the given theorem of T L R + TI(lev). In chapter XI, we shall give a finer proof-theoretic interpretation.

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CHAPTER 11

PROOF-THEORETIC INVESTIGATION OF FINITELY ITERATED REFLECTIVE TRUTH w w w

w w

~57. w

The ramified system RS n Cut elimination Some derivable sequents of RS n Embedding I T ~ into RS n The upper bound theorem for I T ~ Upper bound theorems for TLR and its subsystems Conclusion: the conservation theorems Appendix: primitive recursive cut elimination for RS n

In chapter X we embedded the theory of truth with levels into infinitary systems I T ~ with iterated truth predicates, where level variables and level quantifiers are explained away. We proceed further ahead in investigating the arithmetical content of the systems I T ~ and TLR. We first design an infinitary system RS n in which T n is split into a family {Tna'a < F0} of approximations. The Tna's are linked together by natural recursive conditions, which can be encoded by symmetric introduction rules with the cut elimination property (w167 Then we embed I T ~ into RS n by a modified version of the asymmetric interpretation technique of w (see w167 54-55). The analysis of cut-free RS nderivations readily implies that RSn-theorems of level< n are already derivable without Tn-rules , of course at cost of greatly increasing the derivation length. This increase is given an upper bound with the main theorems of w167 56-57. As a final step, if we formalize the whole reduction procedure, we realize that the proof-theoretic analysis only involves OP-principles, except for the schema T I ( < Fo) of transfinite induction for operational formulas along each a < F 0. We can conclude-with the help of the main result of Ch.VIII-that the operational consequences of TLR are already consequences of OP + T I ( < F0). Similar results hold for the systems MF, MFp, MF c of Ch. II. In particular, the operational consequences of MF (MFp, MFc) are axiomatized by OF + T I ( < r (OP + T I ( < Cw0), OF respectively).

Proof Theory of Finitely Iterated Reflective Truth

312

[Ch.ll

52. The ramified system llS= To a certain extent, the system RS n is meant to be a constructive simulation of the n-th stage in the recursion-theoretic model of Ch.VIII (see w The truth predicates of level < n are assumed as given and satisfy closure conditions corresponding to the IT~-axioms for the predicates T i and F i with i < n; on the contrary, T n and F n are built up in stages (here ordinals < Fo) , and there are rules for passing from stage a to stage a + l and for collecting information at limit stages )~. The essential fact is that the corresponding rules can be symmetrically arranged as introduction rules for T a+l and -~T a+l (similarly for F a + l and -~Fa+l). As a consequence, the standard predicative cut elimination procedure of Schfitte applies, and this property grants elimination of level n statements from derivations, whose conclusion refers only to lower levels. As the reader will see, each RS n is an infinitary calculus; however, it is well-known how to represent infinite derivations by suitable finite data structures (Mints 1975, Schwichtenberg 1977, Buchholz 1991). It follows that the cut elimination procedure is indeed primitive recursive and "tractable", within a fragment of OP, possibly expanded by transfinite induction principles. 52.1. The syntax of RS n. The ramified language Ln, r of RS n contains the following primitive symbols" (i) individual variables x0, Zl, z2... ( Z, y, z are metavariables); m

(ii) individual constants 0, SUC, P R E D , P A I R , L E F T , R I G H T , D; the function symbol for application Ap (binary); (iii) predicate symbols True n and Falsen, - , N(unary);

Vr, FI (binary) and

(iv) level constants for each level < n (i, j, k syntactical variables);

(v) (vi)

ordinal constants ~, for each ordinal a < F 0 ; the logical constants V, A, --, V, 3.

52.1.1. The L-terms are exactly the L-constants; the o-terms are exactly the ordinal constants; i, j, k range over L-terms, while we ambiguously use lower case Greek letters c~, /~, 7, e t c . . . , both for o-terms and the corresponding ordinals. Ln, r -terms are inductively generated from individual constants and variables by use of applications; thus they coincide with the terms of the underlying combinatory logic, and we stick to the previous conv~n~ion~ and definitions. Expressions of the form t - s, - ~ t - s, Nt, -~Nt are called e-atoms, while Yr(i, t), -~Yr(i, t), Fl(i, t), -~Fl(i, t), Truen(t, ~), -~Truen(t, ~), Falsen(t , ~), -~Falsen(t , ~) are called T-atoms

The Ramit~ed System R5 n

XI.52]

313

(t, s range over individual terms, i is an L-term and a is an o-term). s are inductively generated from s by use of the logical constants A, V, V, 3. It is understood that we adopt the variable separation property VSP of w We also adopt the more perspicuous abbreviations Tit "-- Yr(i, t) and Fit "- Fl(i, t), while T~(t) "- Truen(t , ~) and Fan(t)"- Falsen(t, ~). We generally omit the subscript n and we simply write Ta(t) and Fa(t), whenever n is clear from the context. Similarly t~l% := Ta(st) and t~as := Fa(st). NB. The m a p A ~ [A] is left unchanged (see 50.1.2); so it is well defined for Ln-formulas and formulas of Ln, r not containing the new predicates T ~ and F~; the expression {x:xrl~y} does not make sense. 52.2. If A E Ln, r, L c ( A ) ( - the logical complexity of A) is the number of logical symbols occurring in A (-~ excluded). 52.3. Level and n-stage o f A (A formula s (i) (ii)

Let o -

A, V, Q -

V, 3.

Lev(Tjt) = j and Lev(Tan) - n ; i f A is an e-atom, Lev(A) - 0; Lev(B o C) - max(Lev(B),Lev(C)) and Lev(QxB) - Lev(B). Stn(A ) = 0, if A is an e-atom or Lev(A) < n; Stn(A ) = ~, if A is a T - a t o m of level n and ~ occurs in A; Stn(B o C) = max{Stn(B),Stn(C)} and Stn(QxB ) = Stn(B ).

52.4. n-rank of A (A formula of Ln, r)"

Rn(A ) = 0, if n = 0 and A is an e-atom or Lev(A) < n; else: n.(T

t) =

Rn(B o C ) =

=

max{Rn(B), R n ( C ) ) + I and Rn(QxB ) = Rn(B)+I.

52.5. The a-transform Aa of a formula A E s is the Ln, r-formula, which is obtained from A by replacing each occurrence of the atoms (~)Tnt , (~)Fnt respectively with (-~)Tat, (-~)Fat respectively. Similarly, the c~-transform of A E L (L = language of the basic systems of Ch. II) is obtained from A by replacing each occurrence of (-~)Tt in A with (-~)Tat. 52.6. L E M M A (i)

If A is a formula of Ln, r, Rn(A) - w. S t n ( A ) + m , for some mew;

(ii)

If A is a s.~-fo~mut~ ( o~ a formula of L), Rn(Aa) < w ( a + l ) .

Before specifying axioms and rules of the new system, we define, in analogy

Proof Theory of Finitely Iterated Reflective Truth

314

[Ch.ll

with 50.1.6" 52.7 (i) T%Clause(t) := 3x3y((t = [x = y] A x = y) V (t = [Nx] A g x ) V V ((t - [Tx] V t - [Tnx]) A T a x ) V ( U {t - [Tjx] A T j x " j < n}) V V (t = (~x) A Fax) V (t = (x A y) A T a x A Tay) V (t = (Vx) A Vv. Ta(xv))). (ii) F%Clause(t) := 3x3y((t = Ix = y] A-~x = y ) V (t = [gx] A-~Nx) V V ((t - [Tx] V (t - [Tnx]) A Fax) V ( U {t - [Tjx] A ~ T j x " j < n}) V V (t = (~x) A T a x ) V (t = (x A y) A (Fax V Fay)) V (t = (Vx) A 3v. Fa(xv))).

RS n is a Tait-style sequent calculus, like the systems STLR and I T ~ of w167 so, we have to specify axioms in sequent form and introduction rules for the logical and mathematical primitives. 52.8. Axioms of RS n. We assume the substitution closure of the following sets of sequents. 52.8.1. LOG ( - Logical axioms)" (i) t - - t; (ii) - ~ t - s, ~A(t), A(s) ( R n ( A ) - 0 ) ; (iii) -~A, A ( R n ( A ) - 0 ) . 52.8.2. O P E R ( - Operational axioms): (i) (ii) (iii) (iv) (v) (vi) (vii)

K a b - a and S a b c - ac(bc); - ~ C - C' (C, C' distinct individual constants); ((al, a 2 ) ) i - ai, where i - 1, 2; D-~ ~ c d - c; Dk -~ c d - d; -~(~+1) - 0 ; P R E D ( - ~ + I ) - -~. Proviso: k, ~ stand for distinct numerals.

52.8.3. P E R S i j 52.8.4. C O N S i ( -

( - Persistence)" -~Tia , Tja, for i < j < n; Consistency)" -,Tie , -~Fia (i < n);

52.8.5. F I X i ( - Fixed point axioms for level i < n): (i) -~Ti-Clause*(a), Tia and-~Tia , Ti-Clause*(a); (ii) -,Fi-Clause*(a), Fia and-~Fia , Fi-Clause*(a ).

The Ramified System R5 n

XI.52]

Ti-Clause*(t),

315

Fi-Clause*(t ) are defined in the previous chapter (see

w 52.8.6. I N I n ( - initial Tn-axioms): -~T~ and -~F~ if n - 0; else, if n > 0:

-~T~ ,Tn_ I t ; -~F~ , Fn_ it; -~Tn_lt , TOt; -~Fn_lt , F ~ NB: the level terms occurring in the axioms are all < n; if n - O , C O N S i , F I X i , P E R S i j must be omitted, for i < n. Every formula occurring in the axioms has n-rank 0. 52.9. Rules of RS n. They include: (i) the standard logical inferences ( A ), ( V ) , (Vx), (3x), (Cut); (ii) the N-rules: (-~N) " " F ' - ' t - ~ " "

(for e a c h m E w )

F,-~Nt

F, t - m ( f o r s o m e m E w )

; (N)

F, N t

(iii) Successor rules for T n and F n (remind that (Ta+l)

F, Ta-Clause(t)

F, Ta+lt

;

(Fa+l) F, Fa-Clause(t) F, Fa+lt

T a, F ~ stand for Tn~,

(-~Ta+l)

F,-~Ta-Clause(t) F, ~ T a + l t

(~Fa+l)

F,-~Fa-Clause(t)

;

;

F, -~F~+lt

(iv) Limit rules for T n. Let c~ < F 0 be a limit:

(T_LIMa)

F, Tf3s F, Tas ' for some ~ < a;

(-~T-LIM a)

F, -~T ~ s . . .

F, ~Tas for every fl < a.

(F_LIMa)

F, Ff3s

F, Fas ' for some fl < c~; ( ~ F - L I M a)

F, ~ F f~s...

F,-~Fas for every fl < c~.

The rules and axioms of RS n induce a relation of RSn-derivability for finite sequents F of s Remind that low Greek letters a, fl,... range over arbitrary ordinals < Fo, but also over ordinal constants of RS n.

Proof Theory of Finitely Iterated Reflective Truth

316

[Ch.ll

52.10. Inductive definition of the derivability relation RS n t- Olp F (n E w). DER.1. If (the finite set) F _~ F' and F' is an axiom of RSn, RS n F- pOt F, for every c~ and every p; DER.2. Assume: (i) (ii)

RS n F- pZ F~3, for every t3 < 6; F follows from {F~"/3 < 6} by means of the rule ~, where :1 is an inference of RS n with 6 premises (0 < 6 < F0);

(iii)

%3 < c~ for every/3 < 6;

(iv)

sup{p~'/3 < 6} < p and p < p, where # "- R n ( A ) + I , if ~is a cut with cut formula A; else p := 0. Then RS n F- C~l-," p

The previous inductive definition immediately implies: 52.11. L E M M A (Monotonicity of ordinal assignments).

If RS n ~ ~ F and cr < fl p < 6, then RS n F - ~ F . 52.12. N O T A T I O N (i) (ii)

RS n F- ap F "- F is RSn-derivable with length _< c~ and cut rank < p. RS nF- < po~ F . - R S n F - ~ r , f o r s o m e / 3 < a a n d s o m e 6 < p .

w53. Cut d i m i n a t i o n Following the classical method of Schfitte, we show that every sequent derivable in RS n is already RSn-derivable with cut rank at most 1. As a preliminary step, we collect a few simple properties of the derivability relation for RS n. 53.1. L E M M A (i) (ii)

Weakening: if RS n ~ ogp F, then RS n t- ~p F,A. Substitution: if RS n F- p r(a), then R S . ~- p r [ a " - t]

P R O O F : by induction on c~. (i) is a consequence of clause DER.1 of the definition of derivability for RSn; (ii)follows from the fact that RSn-axioms are closed under substitution. !-1

Cut Elimination

XI.53]

317

53.2. D E F I N I T I O N (i) A formula A is reducible to A iff one of the following conditions holds: 1. A = B A G and A = {B} or A = {C}; 2. A = B V C and A = {B,C};

3. A = VxB and A = {B[x := t]}, for some t free for x in B; 4. A -

(~) T~+l(t) and A -

{(-)T~-Clause(t)};

5. A -

(-~)F~+i(t) and A -

{(-,)F%Clause(t)};

6. A - - ~ T ~ t (-~F~t), a limit and A -

{-~T~t} ( A - {-~F~t}), for

some ~ < a. (ii)

A is reducible iff A is reducible to some A.

Clearly, a formula of RS n is reducible iff it can occur as active formula in one of the following inferences: ( A ) , ( V ) , (V), ((-~)T~+I), ((-~)F~+I), (-~T-LIM~), (-~F-LIM~), (fl limit). 53.3. LEMMA.

If Rn(A ) > 0 ,

A is reducible to A

and B E A,

then

Rn(B) < Rn(A). The verification is obvious by definition of n-rank (52.4). 53.4. LEMMA (Inversion). Let RS n Fpa F, A with Rn(A ) > 0 and let A be

reducible to A. Then R S n F pa F , A. P R O O F . Induction on c~. Case 1: F,A is an axiom. Since no reducible formula with n-rank > 0 is active in the axioms of RSn, F, A is still an axiom. Case 2: A = YxB and A is active in the inference ~ = (V) which concludes to F,A. Then we must have, possibly by use of 52.11 and weakening lemma, RS n F p~ F, YxB, B(a) (where a is an eigenparameter not in F,A), for some /3 < c~. Moreover A has the form B[x:= t], for some t. Then by IH, RS n F p~F, B(t), B(a), whence RS n F ~ F, B[x "- t] by the substitution lemma 53.1 (ii) and monotonicity. Case 3: A is active in the inference 5 which concludes to F, A, but ~ ~ (V). Then F, A follows by applying IH to the premise of F, A, which is obviously determined by the given reduction A. Case 4: A is not active in the inference :J which concludes to F, A. Then by IH we can replace every occurrence of A in the premises of ~ by means of A and finally conclude with ~. FI

Proof Theory of Finitely Iterated Reflective Truth

318

[Ch.ll

We proceed to the crucial step in the proof of cut elimination; but we first need the notion of natural ordinal sum. We know that, by the Cantor normal form theorem 45.3, every ordinal 7 is uniquely representable in the form 7 1 + . . . +Tn, with 71 >--'" >--7n, where each 7i has the form J ( i ) f o r some ~(i). If O~1 + . . . -t-O~k and c~k+1 + . . . +C~k+m are the normal forms of c~ and fl respectively, we define:

O~#fl "-- Cr

) -I-...-[-

O~Tr(kTm),

7r being the permutation of { 1 , . . . , k + m } such that a,rll ) >_ ... _> .a,r(k+m ). Clearly # is commutative and strictly increasing in each variable: 1.e. o < 7 implies both a:]/:6 < a # 7 and 6 # a < ")'#a. 53.5. LEMMA (Reduction)

q rts

O~

r,a

RSn F A, A

>_ 1,

P R O O F . We argue by induction on c~#f~. Case 1" I', A and A,-~A are axioms. Then F, A is already an axiom because neither A nor ~A can be active formulas of an axiom, having n-rank > 0. Case 2: We may assume that, say, I',A is not an axiom. (This is not restrictive: since Rn(A ) -Rn(-~A), the whole argument is symmetric with respect to F, A and A,-~A}. 2.1" A is not active in the application of ~, ~ being the rule applied to infer F, A. Then ~ has the form:

Pk F', A, Bk,... ~ for each k C (0, 5), infer RS n ~- pa F, A from RS n t- ak where 5 is the number of the premises of ~. Since ~k#fl < ~=]/=fl, we get by IH, for every k E (0, 6), RS n [-- ; k~/3 r', Bk, A. The conclusion follows by application of 1t with length a k # f l # l 0, "r > / ~ + wk and put a n "- ~+wkO(n+l).

We check, by secondary induction on n E w:

RS o F w-), < ~(-r+l)r[fl,7], ~rlana.

(2.1)

Indeed, by main II-I we also have, for arbitrary fl > 0 and a(fl) - fl+wk~

,-Oqa(~)a

RS 0 b :{a(~)+l

(2.2)

RS o b :/aa/~/++~/F[fl, a(fl)],-~tq~a,(t+l)r]a(~)a (t arbitrary).

(2.3)

If n - - 0 , (2.1) is simply a consequence of (2.2) with the persistence lemma 55.4 and 7 > ao. If n -- m + l , we have by secondary IH: RS 0 F < w(~,+l)rift,,),] ' m~/ama.

(2.4)

If we choose ~ = am in (2.3) above, we get, since a(am) = an: RS o I-- w(O-n+l W(an+l I

a"

~qana.

(2.5)

-~rlana.

(2.6)

But 7 > an and hence, by persistence lemma 55.4, RS 0 I-- w-'/ < ~(o'-F1) r[fl, 7] '

~rlama,

An application of (Cut) between (2.4) and (2.6) concludes the verification of (2.1). On the other hand, if we apply the substitution lemma, we obtain, with t arbitrary, for each n: a.

(2 7)

r[fl, ~/]' "~t - -~, trl'la "

(2.8)

RS 0 }_ w')' < w(-/+l) r [ z ,

~

-

Hence by 54.3 (iii) and a cut of rank wa n < wT: RS 0 ~ toy < w(-/+l)

A final application of (-~N) yields (2.8): w("/+l

RS o k w(~/+l / r[fl, "y], -~Nt, tri~a. 0 Finally, if MFp b A, we have for some n > 0, by (1)-(2) with/~ - 1:

Upper Bound for TLR and its Subsystems

XI.57]

333

wn+2

RS o ~ wn+2 A[1, wn], whence by Tait's cut elimination RS o f-r1 A[1, wn]. If we apply the upper bound theorem and the closure properties of Cw0, we obtain O P ~ F - a< r

A, if A E s

and O(CTM, Cw0)l=A, if A is T-positive. !"1

Ad(iii)-(iv). We introduce a variant ITo(c ) of ITo(P) such that, if MF c F- A, then ITo(c ) ~ A. IT0(c ) is obtained from IT0(P) by replacing the rule P-WI N D with C L - N - I N D below: B

infer r, ~Nt, trla from the premises r, Cl(a); r, 0~a; F, Vx(x~a~(x+l)~a). k As for ITo(P), ITo(c ) enjoys partial cut elimination, i.e. if ITo(c ) F- m+aA,

then ITo(c ) F- 1n A, for some n E w. Instead of reproving the appropriate form of embedding for ITo(c), we establish the crucial case of the interpretation theorem in semantical terms; we then give directions to obtain its proof-theoretic version. First of all, we recursively define: I=A[m, n] iff either A is an e-atom and CTMI=A; or

(3)

A - Tt (Ft) and t E O(CTM, n) ((-~t)E O(CTM, n)); or A - - ~ T t (~Ft) and t ~ O(CTM, m) ((-~t) ~ O(CTM, m)); or A - VxB (3xB) and ]=B(t)[m, n] for every (some) t E CTM; or A-

B A C (B V C) and [=B[m,n] and (or)I--C[m,n].

We also write [= Tnt (Fur) for I= (Tt)[m, n] (I= (Ft)[m, n]) and I= ~Tmt (~Fmt) for I= (-~Tt)[m, n] (]- (-~Ft)[m, n]). If F - {A1,... , Ak} , I= F[m, n] is interpreted disjunctively as I-- (A1 V ... V An)[m,n ]. As expected, we have: if I=F[m ', n'],A and m < m' < n' < n, then I=F[m, n], A.

(4)

Now we claim: if IWo(c ) F- k F, then I=F[m, n], for every m > O, n >_m+2 k.

(5)

Verification of (5): by induction on k. Let us only consider the case where, for some/c o < k, we have: ko

~

ITo(c ) t- 1 F Cl(a)

ko

ITo(c ) F- 1 F, Or/a

and

(5.1)

ko

ITo(c) F- a -~t71a,(t+ 1)rla (t arbitrary). Then by I n applied to (5.1), and (4), we have, for every m > 0, n > m + 2 ko with Po - m+2k0 and remembering that t71P~ - TP~

t~P~ - FP~

9

334

Proof Theory of Finitely Iterated Reflective Truth

[Ch.ll

I=F[m, n], to po a, t~ p~ a (t arbitrary);

(5.2)

I=r[m, nl, OnPOa;

(5.3)

I=r[m,n],-~to m a, (t+l)n po a (t arbitrary).

(5.4)

Now fix m I > 0, n 1 > m 1-4-2k and set Pl - m l +2k~ We check by secondary induction on l E w I=F[ml, nil, I~Pl a.

(5.5)

If I - 0, apply (5.3) with m - m 1. If I - j + l , assume by secondary IH:

(5.6)

I-F[m 1, n 11, ~r/pl a. If we apply (5.4) with m -

Pl, we get, for q - Pl+2k~

(5.7)

I=F[pl, q], ~ ~//Pl a, lT]q a. By (4), since n 1 > m 1+2 k > Pl +2k~ - q > ml, we get

I=r[ml, n 1], -~ jOP~ a, 70q a; hence, with a cut:

[=F[ml, nll,7~lq a.

(5.8)

By Ch. II, 7.7, we have O(CTM, n) f-! {t" t E CTM, O(CTM, n)l=T~t) - 0 and O(CTM, n) C_ O(CTM, m) for n < m, whence [=F[ml, n l ] , ~ l~q a, --1 lr/q a.

(5.9)

I=F[ml, nl], -~ 7~Pla, 7~qa.

(5.10)

The conclusion follows by application of the cut rule to (5.S), (5.9), (5.10) and (5.2) (in the last case choose m - ml). El If we inspect the verification of (5), we can observe: the levels involved are

finite, by contrast with the corresponding step (2) for ITo(P); the levels depend only on the given parameter m and on the derivation length. Hence the interpretation theorem for ITo(c ) can be carried out within the fragment of RS o with finite levels only. In particular, we can modify the notion of 0-rank for formulas of RSo, in such a way that the 0-rank is always finite and cut elimination still works with respect to the new notion (Hint: choose Ro(T k) - R o ( F k) - 3 0 k and reprove 54.1-5, 55.4). Finally the embedding theorem 55.5 is refined for ITo(c), to the extent that: if IT0(c ) F- ~ A, then RS o F- ~ All, 1+2 k] for some )~ < Co, which implies (iii)-(iv). We leave a complete check of (6) as exercise. [i

(6)

The Conservation Theorems

XI.58]

335

w58. Conclusion: the conservation theorems If we piece together the results of Ch. IX with the main theorems of w57, we can obtain a characterization of the operational theorems of TLR-4-TI(lev), MF, MFp, MF c. In order to state the theorem, let us agree that: (i) a < fl stands for the Lop-formula defining the primitive recursive well-ordering of type F 0 and lower case Greek letters a, fl, 7 . . . r a n g e over the field of < ; (ii) Progr( < , B ) = Vc~(Vfl(fl < c~~ B(fl))--, B(c~)); (iii) TI(c~) is the schema Progr( < , B ) ~ Vfl < c~. B(/~); TI(
L(g), p > R(g), then

(((T-LIMh),a); F, Tht; /3; p; PAR(g); g)E CODE; (xiv)

the clauses corresponding to the introduction of F 6, for 5 limit, are obtained from (xiii) by replacing T with F.

NB" that only derivations with a finite set of parameters are encoded, as implied by (ix), is not restrictive for our aim of embedding finitary systems into infinitary ones. By generalized recursion theory, CODE is the fixed point of a positive arithmetical operator (indeed a boolean combination of II O- and E ~ conditions) and it is generally not elementarily definable. However, we can effectively associate to each derivation label f a well-founded tree of formal figures, which is locally correct (hence a derivation in the true sense), exactly when f E CODE. Since local correctness is an elementary condition, we can get an elementary representation of CODE. 3. D E F I N I T I O N . Let 9 denote concatenation on OT-elements. (i) OT* is the smallest set X such that" ( ) E X; if s E X, s , a E X, for

aEOT. (iN) Length(( ) ) - 0; Length(s,a)- Length(s)+l. (iii) If s, s' E OT*, s C_1s' iff s - s' or s ' - s , a , for some a E OT. C_ is the transitive closure of C_ 1; s C_ s' iff s is a subsequence of s'. (iv) S COT* is an OT*-tree if ( ) E S and S is closed under the subsequence relation C (s' C s E S implies s' E S).

Proof Theory of Finitely Iterated Reflective Truth

342

[Ch.ll

An OT*-tree S is well-founded if every S-path is finite; by S-path we mean a maximal C-chain. 4. D E F I N I T I O N (by course of value recursion on Length(s), s E OT*).

1. D e r ( f , s) - 0, if f ~ L A B E L or s ~ OT*; 2. else, assume f E L A B E L , s E OT*: then 2.1. Der( f , ( ) ) 2.2. let s -

f;

s',fl, for some fl E OT; assume ~1- RF(Der(f,s'));

2.2.1. ~ - AX: D e r ( f , s ) -

0;

2.2.2. ~ - REP~: then Der(f, s) - 0 if fl 5/: t~; else, Der(f,s) - Uni(SDl(Der(f,s')),fl); 2.2.3. ~ has ~ premises ~ > w, 6 limit: then Der(f, s) - 0 if fl > 5; else,

D e r ( f , s ) - Uni(SDl(Der(f,s')),fl ) (here Uni is a fixed recursive universal function for the unary PR-functions); 2.2.4. ~ I - ( V ,i), i - 1 or 2: then D e r ( f , s ) fl - i - 1; else Der(f, s) - 0;

SDi(Der(f,s'))if

2.2.5. ~ is not as in 2.2.4, but is k-ary (1 _< k _< 2). Then D e r ( f , s ) - 0 if fl > k; else n e r ( f , s) - SD~+ l ( n e r ( f , s')). By inspection of 4, we obtain: 5. L E M M A (i) The operation )~x)~y.Der(x,y) is primitive recursive in Uni ( = the given universal function for 1-ary PR-functions). (ii) We can find a PR-operation D E R such that D E R ( f ) is an index for )~x. Der( f , x). Clearly (ii) follows by the s-m-n theorem. Henceforth we write s E f for D e r ( f ,s) # O and we say that s i s a n o d e o f f . T ( f ) - {s E OT*" s E f } is obviously an OT*-tree. 6. D E F I N I T I O N . L C ( D E R ( f ) ) holds iff it satisfies the following conditions:

1. f E L A B E L , ( ) E f; D e r ( f , s ) E L A B E L , if s E f; 2. if s E f, ~ = R F ( D e r ( f , s ) ) , then 2.1. either ~ = A X , E g D ( D e r ( f ,s)) is an axiom and s.~ ~ f for every ~ E OT; or 2.2. ~ ~= A X , E N D ( D e r ( f , s)) follows from the set

Appendix

XI.A]

343

{ E N D ( D e r ( f , s , f l ) ) : s,fl E f} by means of 3; 3. in 2.1-2.2 the conditions on the length and the rank are met; this means in particular that if s E f, s,/3 E f, then L(Der(f, s)) > L(Der(f, s,j3)). { L C ( D E R ( f ) ) is read as " D E R ( f ) encodes a locally correct derivation"}. 7. LEMMA. Let f E L A B E L . (i)

The predicate L C ( D E R ( - ) ) is definable in s

(ii)

L C ( D E R ( f ) ) implies that T ( f ) is well-founded;

(iii) f E C O D E iff L C ( D E R ( f ) ) holds. PROOF. (i): by a straightforward formalization of 6. (ii): by condition 6.3, any descending infinite sequence in the tree ordering would produce a descending infinite sequence in OT*. (iii): ~ : by generalized induction on the definition of CODE. ~ : by (ii), T ( f ) is well-founded and we can apply transfinite induction to verify

f E CODE. D As consequence of 7 (iii), there is an s A(f), which represents the condition f E CODE; for convenience, we write f E D E R instead of A(f), and we simply say that f is a derivation. Lemma 7 makes possible to define elementary operations on CODE; more properly, when we talk about operations on DER, we mean PR-operations

F : LABEL---, LABEL, which preserve the property CODE, i.e. if f E CODE, F ( f ) E CODE. As a rule, we only rely on the surface structure of labels, without appealing to transfinite induction; instead, we apply course-of-value recursion (on the depth of the derivation labels), PR-distinction of cases and the second recursion theorem for PR-indices (after Kleene 1958): 8. THEOREM. There is a Kalmar elementary operation Fix such that, if e

is a PR-index for a k+l-ary PR-function, then Fix(e)E P R I and [Fix(e)](nl, . . . , n k ) = [e](Fix(e), ha,... , nk). Operations on DER and cut elimination We essentially exploit the finitary presentation of RSn-derivations; the basic operations involved in cut elimination are shown to be primitive recursive and they naturally work on arbitrary derivation labels. Transfinite induction is needed only for checking that the basic operators work properly (correctness proofs). First of all, we see that it is possible to deform

Proof Theory of Finitely Iterated Reflective Truth

344

[Ch.ll

monotonically the ordinal assignment (of lengths and ranks), and to rename parameters. If f E L A B E L , f(a) abbreviates the condition "a E P A R ( f ) " ; f ( a / t ) denotes the result of replacing everywhere a by t. D E R is not closed under the operation t~-. f ( a / t ) , because substitution may spoil the eigenparameter restrictions. In order to reconcile the present notations with those of w52, we introduce the following 9. DEFINITION. f F- aP F "- L C ( D E R ( f ) ) A f E L A B E L A

A E N D ( f ) = F A L ( f ) = a A R ( f ) = p. Hence, as a consequence of lemma 7, we have:

f F - pa F is definable in the language s

9.1

Now we state a few technical lemmata, which are essential for manipulating derivations. Proofs are straightforward: however the formal definitions of the primitive recursive operations, which are claimed to exist, require lengthy PR-distinction by cases, course-of-value recursion on D E P and a final application of the second recursion theorem. The technique will be illustrated below in the case of the inversion lemma and the subsequent theorems. 10. LEMMA (i)

We can find a PR-operation M O N such that, if f F- pc~

F~ o~ _ < C~~,

Ol !

p 1:

C(a,f)-(:1; F; Cpc~; 1; V; C(a, fo); C(a, fl) ) (this is well-defined since

D E P ( f i ) < DEP(f), i C {0,1}).

3.2. ~ is infinitary, so f0 is a PR-index: we let

C ( a , f ) - (:t; F; Cpc~; 1; V; comb(a, fo)), where

comb(a, fo) is a P R - i n d e x for Sx. [a]([fo](X)).

3.3. ~ is a cut of rank v - tJ(f) > 1, v < p and L(fi) - ai, for i - 0,1. By lemma 11, there is a PR-function #i such that #i(a) satisfies [#i(a)](f)- MOg([a](fi),r v), whenever a E PRI. By lemma 14, there is a PR-function h such that, for a E PRI:

[h(a)](f) - [a](RDC([Po(a)](f), [Pl(a)](f))). Hence, recalling lemma 13 (item 5) and s-m-n theorem, there is a PRfunction r such that, if a E PRI: [r

f)](x) --

[O]([h(a)](f),x).

Finally, we put:

C ( a , f ) - (REPo; F; Cpc~; 1; r If we choose a PR-index CF by theorem 8, we can verify, by main induction on p and secondary induction on a, that CF(f)F- r F. [:] W i t h similar arguments, we can prove that there is a P R - o p e r a t i o n satisfying Tait's second cut elimination theorem (and hence 1-step cut elimination). By inspecting the construction of theorem 15, we can conclude that OP+TIop(Fo) proves the cut elimination theorem for RS n.

PART E

ALTERNATIVE VIEWS

"On a signal~ beaucoup d'antinomies, et le d~saccord a subsistS, personne n'a ~t~ convaincu; d' une conlradiction, on peul toujours se tirer par un coup de pouce ! Je veux dire par un distinguo." (H. Poincar4, 1913).

This Page Intentionally Left Blank

CHAPTER 12

NON-REDUCTIVE SYSTEMS FOR TYPE-FREE ABSTRACTION AND TRUTH w w

w w w

~64.

The core system V F - and transfinite induction Supervaluation models of V F An abstract sequent calculus and truth Cut elimination and related properties A provability interpretation and the upper bound theorem Reconciling supervaluation models with provability interpretation

In this chapter we critically reconsider the basic truth axioms of w7 and their semantics. An essential feature of these principles is that they are reductive: they (roughly) presuppose a "compositional theory of meaning", in that the truth conditions of logically complex sentences are reduced to corresponding conditions for logically simpler sentences. In particular, the basic idea is predicativistic in spirit: a statement is justified only if its truth can be ultimately grounded upon elementary truths (see Kripke' s classification in w34). A major consequence of this general attitude is that even a tautology may not be accepted, whenever it involves ungrounded sentences. It is therefore natural to investigate the consistency of alternative views, which are well-behaved with respect to logical consequence. In particular, it seems reasonable to accept every classical tautology A, irrespective of its complexity and its specific content (e.g. A might have the form r~lr ~ r~r, r being the Russell property). In w we present a non-reductive system VF-, which has non-trivial mathematical content (indeed, it proves a generalized induction principle). The main theorem tells us that VF-, even if number-theoretic induction for classes is assumed, is proof-theoretically equivalent to OP (and hence to PA); furthermore, the same equivalence holds between VFp "- "VF- plus internal number-theoretic induction axiom" and the impredicative theory ID 1 of elementary inductive definition. Thus the non-reductive approach overcomes the deductive limits of the reductive notion of reflective truth. The rest of the chapter is devoted to illustrate two types of semantics for VF-: supervaluation models (SV-models, in short) and a provability interpretation. SV-models are introduced in w and they take inspiration

Non-Reductive Systems for Truth and Abstraction

352

[Ch.12

from the supervaluation method (van Fraassen 1968, 1970). Indeed, we show that there is a simple monotone operator, whose fixed points provide models for VF-. w167 describe a proof-theoretic semantics for VF-: the truth predicate T is interpreted as provability in an abstract infinitary system, which enjoys cut elimination (w We underline that the proof-theoretic investigation is worked out in a restricted metatheory, i.e. the theory P W - + GID of w16 with approximation operator and generalized induction. As a byproduct, we shall obtain an upper bound for V F - and, at the same time, insights on new principles for truth (w In addition, w shows that for countable ground structures, SV-least fixed point models and provability models coincide.

w59. The core system V F - and tran.,fflnite induction

For the sake of simplicity, we restrict our consideration to a variant of .5, which assumes ---~, V as primitive logical symbols; we let _L " - ( K - S ) and -~A "-(A---, _L ), while V, A, 3 are defined as usual in classical logic. We write T A and FA for T[A] and T[-~A], [A] being the term__ of .5op, which represents A (cf.w we assume that I M P L Y "-AxAy.(31,(x,y)) encodes ---,). In order to simplify a few arguments below, it is convenient to fix an axiomatization of classical logic with modus ponens as the only inference rule (Tarski 1965). 59.1. DEFINITION (i) V F - is the elementary theory (in the language s as modified above), which includes classical first-order logic with equality, the axioms of O P - and the five schemata below:

T-out:

T A ~ A (A arbitrary sentence);

T-elem"

(A---, TA), where A has the form t = s, -~t = s, Nt, ~Nt;

T-imp:

T(A~B)~(TA~TB);

T-univ:

VxTA~TVxA;

T-log:

TA, provided A is a logical axiom.

(ii) VF c " - V F -

plus the class number-theoretic induction axiom CL-

N I N D , i.e. the formula Cl(a) A Orla A Vx(xrla ~ (x+l)r/a) ~ Vx(Nx ~ xrla); VFp "- V F - + P - N I N D "- Orla A Vx(xTla ---, ( x + l ) r / a ) ~ Vx(Nx --, x~la);

P - N I N D is the number-theoretic induction for properties.

XII.59]

The Core System and Transfinite Induction

353

59.1.1. REMARK. The T-schemata above are theorems of NMF-; what really makes the difference, is the closure of T under logical deduction. 59.2. LEMMA (i) The LOG-rule: if A is a formula of 2. and A is provable in pure logic, then V F - F TA. (ii)

The internal abstraction schema: VF- F T(Vu(u~{x: A} ~ TA[x := u]));

(iii) VF- F T A A T B ~ T ( A A B); (iv) VF- F -~(TA A FA); (v) VF- F (TA V F A ) ~ ((A ---,T B ) ~ T(A ---,B)); (vi) VF- t- T A V T B ---,T ( A V B); (vii) VF- F T V x A ~ VxTA; (viii) VF- F 3 x T A ~ T3xA; (ix) VF- F T A ~ F-~A; (x)

if A is a formula of s

(A does sol contain occurrences of T),

VF- F ( T A V FA) A (TA ~ A). PROOF. (i) By induction on the derivation of A in pure logic. If A is a logical axiom, we are done by T-log. If A is obtained from B and B--, A, we get T B and T(B--, A) by IH, whence TB--~ T A by T-imp, and finally TA. (ii) By identity logic and (i) above, we have (u fresh variable):

T((Ax.[A])u = [A[x := u]]---,.(ur]{x : A} ~ TA[x := u])). We then obtain, by T-imp and T-elem,

(~x.[A])u = [A[x := u]] ~ T ( u ~ { x : A} ~ TA[x := u]), whence T(uy{x: A } ~ T A [ x := u]) by /%conversion. The conclusion follows by logic and T-univ. (iii) AAB---,A, A A B ~ B hold by logic, whence by LOG-rule T(AAB)~TAATB; in the other direction we apply the tautology A ~ (B -~ (A A B)) and T-imp. (iv): trivial consequence of T-out. (v) Assume T A V F A , A - , T B . If T A holds, then A holds by T-out, whence TB; but T ( B - ~ ( A - - , B)) by LOG-rule; hence TB--, T ( A ~ B) with T-imp, i.e. T ( A - ~ B ) . If F A is assumed, we have T ( ~ A - - , ( A - ~ B ) ) by LOG-rule; the conclusion again follows by T-imp. The reverse direction

354

Non-Reductive Systems for Truth and Abstraction

[Ch.12

follows by observing that T A V FA implies A ~ T A (use T-imp, T-out). (vi)-(ix): by LOG-rule and T-imp. (x): by induction on A, applying T-elem, LOG-rule and the previous facts. 0 59.2.1. REMARK. (i) T ( T A ~ A ) i s inconsistent with T-out, T-imp and T-log. If we choose A :--"TL, where L - [-"TL], we have the following chain of implications:

T ( T L ~ L) ~ T(-"L ~ -"TL):=V T(-"L ~ L ) ~ TL ~ T-"TL ~ -"TL. (ii) Assume the schema T ( 3 x A ) ~ 3xTA. Then we have in pure logic, with -"K=S, 3x(x=g~A), hence T ( 3 x ( x = g ~ A ) ) , i.e. for some c, T(c=K~A), which implies by 59.2 either ( c = K A T A ) o r (-"c -- K A T-,A), i.e. T A V T-"A: absurd. To sum up, V F - + { T 3 x A ~ 3 x T A } is inconsistent. But a special case of the schema is consistent with V F - (see 63.9). Fix an enumeration {Ai} i ~ ~ of s which have exactly two distinct free variables. Let x 9 y stand for any formula Ai(x , y) in the given list. 59.3. DEFINITION (i) Progr( 9 ,B) := Vx(Yy 9 x.B(y)---, B(x)) ( = the property defined by B is 9 progressive); (ii) W ( 9 := Y z ( P r o g r ( 9 ~xrlz); (Progr( 9 stands Progr( 9 B) with B(x):= xrlz and we simply say that z is 9

for

(iii) WF( 9 ):= {u: W( 9

WF( 9 ) is clearly suggested by the set-theoretic definition of the largest well-founded part of a relation; remarkably, VF-justifies the corresponding transfinite induction schema. 59.4. T H E O R E M (Transfinite induction). VF-proves: (i) Progr ( 9 WF( 9 )); (ii) Progr( -~ , B)-~ Vx(x~WF( -~ )-~ B(x)), where B is an arbitrary s PROOF. (i) Progr( 9 ,W( 9 , - ) ) i s clearly derivable in pure logic; by LOG-rule we can infer

T(Vx(Vy 9 x.W( 9 , y) ~ W( 9 , x))). Since

9 is defined by a formula of s

hence (1)

we have T(x 9 y)V F(x 9 y) by

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355

59.2 (x). If we apply 59.2 (vii), 59.3 and 59.2 (v), (1)implies Vx(Vy -~ x. T W ( -~ , y) ~ T W ( ~ , x)),

(2)

which yields Progr( -~, W F ( -~ )) by means of T-out and 59.2 (ii). (ii) It is enough to check

Vx(x~lWF( -~ )---* B'(x)), where B ' ( x ) " - P r o g r ( -~ , B ) ~ B(x). In pure logic, we have

Progr( -~ , B'). Then we can repeat the argument for (2), thus obtaining

Vx(Vy(y -~ x ~ T B ' ( y ) ) ~ TB'(x)), whence by abstraction,

Progr( ~ , {u" B'(u)}).

(3)

If we assume x~IWF( -~ ) and we choose z "- {u" B'(u)}, we get

Progr( -z,, {u" B ' ( u ) } ) ~ X~l{U" B'(u)}.

(4)

From (3)-(4)and abstraction, it follows TB'(x), hence B'(x) with T-out. F1 59.4.1. REMARK. The previous argument only requires that { ( x , y ) ' x -~ y} is a class, and not the stronger s To appreciate the strength of VF-, the reader with a "logicistic" inclination may be willing to verify the following theorem. Let s be 2. without the predicate N and with the combinators K and S as the only individual symbols; let VF 0 be the subsystem of VF-, formalized in the fragment s Then we have: 59.5. THEOREM. Peano arithmetic is interpretable in VF 0. PROOF (Hint). Simply define (x-~-l)"- {x}, 0 " - 0 and --

Vy(Clo

N(y)-

where ClosN(Y ) "--O~ly A Yu(u71y ~ (u+l)~/y). Plus and times are introduced s la Dedekind as the least relations satisfying the obvious recursive clauses. By adapting 59.4, we can verify the appropriate induction schemata (for details, see Cantini 1991). F1 It will follow from the main result of w63 that VF 0 is not stronger than OP. In contrast to MF, VFp ( - V F - + p r o p e r t y N-induction axiom) goes beyond the limits of predicative mathematics. This is most easily seen by appealing to the theory IDl(acc ) of accessibility elementary inductive definitions, which proves the 1-consistency of Predicative Analysis (see

356

Non-Reductive Systems for Truth and Abstraction

[Ch.12

Buchholz et al., 1981). For the reader's sake, we outline IDl(acc ). If L ( P A ) i s the language of Peano arithmetic, fix an effective enumeration {Ai: i E w} of all L(PA)formulas, containing two distinct free variables. The language of IDl(acc ) is L(PA), expanded by a countable sequence of unary predicate symbols, say {IN: i C w}. As above, let -~ stand for any Ai(x,y); we use WF( ~ )(x) as a more suggestive notation for Ii(x ) (whenever -~ is any Ai). Formulas are inductively generated as usual; atoms obviously have the form t - s, WF( -~ )(t). The axioms of IDl(acc ) contain: (i) Peano axioms; (iN) numbertheoretic induction for the full language; (iii) for each -~ and arbitrary B(x), the axioms:

WF( -~ ).1

Progr( ~ , WF( -~ ));

WF( -~ ).2

Progr( -~ , B)-~ Vx(WF( ~ )(x)-~ B(x)).

59.6. THEOREM. IDl(acc ) is interpretable in VF p" PROOF. By the theorem 59.4, it only remains to check that VF p proves the number-theoretic induction schema for arbitrary formulas of 2.. Set ClosN(A ) "- A(O) A Vx(A(x)--, A ( x + l ) ) and A'(x) "- ClOsN(A ) ---,A(x), where A is a given arbitrary formula of .5; then ClosN(A' ) is trivially derivable in pure logic. Thus V F - ~ TClosN(A'), which implies VF-~-ClosN({X: A'(x)}), whence by property N-induction Vu(iu--,uTl{x: A'(x)}); T-out and exchange of premises imply VFp F- ClosN(A)---~ Vu(Nu--~ A(u)). [-! 59.6.1. REMARK. Conversely, VFp has a model in a set theory, which is proof-theoretically equivalent to ID1; the basic steps are similar to those of the main theorem of w and the result is essentially contained in Cantini (1990). Thus we concentrate on VFc, in accord with our choice of stressing systems not stronger than PA. To conclude, the reader may naturally wonder whether the strength increase sensibly affects the structure of classes in VF-. For instance, does any of the properties WF(-~ ) define a class? The answer is negative and it can be readily obtained with a recursion-theoretic investigation of the inductive models of w By the way, the fundamental closure properties of CL := {x: Cl(x)} in V F - a r e best summarized by the non-surprising 59.7. THEOREM. V F - p r o v e s that CL is closed under the join principle and the elementary comprehension schema (see Ch. II, 9.7-9.9). The proofs are straightforward and make use of the elementary facts of 59.2.

5upervaluation Models

XII.60]

357

w60. Supervaluation models of VFWe keep using the conventions and notations of w7 and w30; we fix a standard operational structure 31~I=OP-; Lop(Mr,) and .5(title) are the usual languages expanded with distinct constants for elements of M, M being the universe of 31,. If tin a closed term of s ,~t~(t)is the value of t i n dtl~. 60.1. DEFINITION (i) Once 3l~ is given, X C_ M and A is an arbitrary sentence of s XI=A stands for "A holds true in the structure ( ~ , X ) " , whenever Lop receives its usual interpretation in Ml~ and T is assigned the subset X, i.e. (all,, X}I= Tt iff 31~(t) E X.

(ii) Xll-A

iff for every Y C_ M, if X C_ Y, then

YI-A (where

A is an

arbitrary s (iii) If X C_ M, we let (I)o(X) : - {3t,([A])'XIJ-A, A L(Ml~)-sentence}. (iv) X is (~o-dense iff X C_ (~0(X); X is (~o-closed iff (~0(X)C_ X; X is consistent (complete) iff for no b E M, b E X and (-,b) E X (for every b E M, b E X or (-~b) E X). CONS(drip)"- {X C_M" X consistent}. (v) SENT(Jft~)"- {~([AI)" A s Since ~ is fixed, we generally omit the explicit indication of 31~ and we simply speak of "sentence" and "consistent" tout court. 60.1.1. REMARK. Variants of the relation X I [ - A are obtained by imposing additional constraints on the possible extensions of X (cf.w67). For instance, if we define the relation X I ] - A by quantifying over all consistent and complete extensions of X, we obtain van Fraassen's notion of supervaluation for s For this reason and because of theorem 60.3 below, the fixed points of the operator (I)0 are also called supervaluation models (in short

SV-models). 60.2. LEMMA (i) X[[- A implies X[=A (A s (ii) Xll- A--, B and XI[- A imply X][- B; (iii) If X[[-A(a) for every a E M, then Xl[-VxA; (iv) (I)0 is monotone: X C_Y =~(~o(X) C_(~o(Y) (X, Y C_M).

Hence FIXo(31~ ) - {X C_M" X -

(~0(X)} is non-empty.

(v) If X is (~o-dense, then X C_SENT(Jf[~) and X]= TA--, A; hence

FIXo(Jf~ ) C_CONS(J~).

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[Ch.12

PROOF" (i)-(iv) are trivial by definition of II- and the Knaster-Tarski theorem. As to (v), if X C_Go(X), T-out holds by (i) and trivially X C_SENT(J~). If NI~([A])E X and ~ ( [ ~ A ] ) E X , then X [ [ - A and X I I - ~ A , which yield a contradiction by (i). E! 60.3. THEOREM (i) If X E F I X o ( ~ ) and ~ I - O P - , then XI=VF-. In addition we have, for arbitrary a E M: T-rep XI= Ta ~ TTa; (ii) If ~ is an w-model of OP and X E FIXo(Jtl~ ), then XI= VFp (cf. 59.1). In particular, if A is an arbitrary instance of N-induction in the language L, XI= TA. PROOF. (i). T-out, T-imp, T-univ: apply 60.2 (v), (ii), (iii). T-elem: if A has the form Nt, t - s or the negation thereof, X[= A implies MI~[=A, whence YI=A, for every Y _DX , i.e. XI[-A, i.e. X[= T A as X is (~o-closed. The converse is similar. T-log: if A is a logical axiom, X[= A, for arbitrary X C M, i.e. X[[-A. Hence, if X E FIXo(Jf6 ), NI~([A]) E X by (I)oclosure, i.e. X]= TA. T-rep: if XI= Ta, also a E Y and Y]= Ta, for every Y 2 X, whence X I I - T a , i.e. X]= T T a ((I)o-closure). (ii) If A is an L(.At~)-instance of N-induction, Y[= A, for every Y C_M. The conclusion follows by Co-closure of X. Vl 60.4. COROLLARY. V F - + T - r e p + { T A " A is a logical axiom or an axiom of OP-, or an arbitrary instance of N-induction in the language s is consistent. At this stage we shall not undertake the investigation of the latticetheoretic structure of the fixed point models of (I)0: suffice it to say that also in the present situation the encoding techniques of Ch. VII can be profitably applied. For instance, the reader can verify: 60.5. THEOREM. Card(FIXo(alg))- 2 card(M). However, we warn against mechanical repetitions of the old arguments.

w61. An abstract sequent calculus and truth We consider the problem of giving a more constructive semantics for VF-; in particular, we show how to avoid universal quantification over arbitrary subsets in the definition 60.1. To this aim, we shall define a generalized sequent calculus %, in such a way that provability in % yields an interpretation of the predicate T of VF-. This step is rewarding in two

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An Abstract Sequent Calculus

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respects. First of all, provability semantics validates new schemata and hence we shall obtain a stronger consistency result. Furthermore, the definition of % and the derivation of its main properties can be easily carried out in the system P W c + GID with N-induction for classes, which is conservative over OP (and hence PA; see Ch. III, 15.5). Since the %-provability interpretation is the identity over s it will follow that VF-, VF c are conservative extensions of OP; in addition, VFp turns out to be interpretable in P W + G I D , the system with full induction schema, which can be shown proof-theoretically equivalent to the theory IDl(acc ) of w The crux of the construction lies in devising a sequent calculus %, which enjoys cut elimination and hence is consistent, provably in P W c + GID. Clearly % has to be infinitary (by axiom T-univ). However, the problem with the usual cut elimination proofs is that they require induction on cut formulas (of maximal complexity), i.e. forms of number-theoretic induction that may not be available in weak systems like P W c + GID. In essence, the solution w e present here is simply to replace the usual finilary sentences

with natural abstract counterparts, which are introduced by generalized induction. It follows that N-induction can be avoided by means of ordinal transfinite induction, which is available in unrestricted form using GID (see w m

Henceforth, we use the abbreviations: (Va)"-t); (VxA)o o = Vx(Aoo ).

If k is a natural number, it is understood that in the context Sentoo(k), k stands for the (closed term representing the) corresponding finite ordinal. 63.2. LEMMA (Local truth). For each finite k, we can define a predicate T R k ( x ) such that P W - + GID proves: (i) T R o ( [ T a ] ) ~ ( F- :=Va);

TRo([X = y ] ) ~ x = y;

TRo([Nx])+-~ g x ;

(ii) TRk+l(a---+b)+--~a~Sentoo(k ) A brlSentoo(k ) A (TRk(a)---, TRk(b));

TRk+I(Va ) ~-~Vx((ax)rlSentoo(k ) A TRk(ax)); (iii) Atoo(a ) ---+(TRk+l(a) ~ TRk(a));

if k < n, arlSent~(k ) ---, (TRk(a) ~-+TRn(a)); (iv) Yx(xrlSentc~(k ) ~ - ~ ( T R k ( x ) A TRk(-,x)); (v) TRk([A(Xl, . . . , Xn] ) +--,A~(Xl, . . . , Xn) ,

for every formula of logical complexity 0, X(n)I= T A V T-,A, i.e. Jfl~([TA V T~A]) E Thoo(J~ ). The case of T-exist is trivial as well. (ii). By lemma 66.2, it suffices to check closure under T-intro and T-elim. Assume 3b([A]) E Thor ). Pick any m > ko(A)+l: then X ( m - 1)]= A and hence X(m)I=TA; so Mt,([TA])E Thoo(J~ ). Thus Thcv(J~ ) is closed under T-intro. Assume J~([TA])EThoo(./tl~) and let m > k o ( T A ) . Were X(m)i=--,A , then .AI,([A])it X ( m + l ) by definition of J, i.e. X(m+I)I=--,TA. But m + l > ko(TA): hence X ( m + I ) I = T A , absurd. It follows that NI,([A])E X(m), for every m > ko(TA), whence .At~([A]) E Thor

).

Thus Thoo(J~ ) is closed under T-elim. [3 66.5. COROLLARY. FSL is consistent, but w-inconsistent. PROOF. FSL is consistent by 66.4 and 66.2 (i); but FSL contains IL and hence it is w-inconsistent by 65.4. [3 66.5.1. REMARK. (i) It is immediate to see that FSL proves the consistency of arithmetical analysis (see 40.3.1). By a result of Halbach (1994), FSL is proof-theoretically equivalent to ramified analysis up to any level < w. (ii) The unrestricted inconsistent, Lhb axiom

Tx---~ TTx.

addition of T T z ~ T x would clearly imply in FSL the Tarski schema T A ~ A (A arbitrary). However, FSL becomes even if we add either T T x ~ Tx or Tx ~ TTx; in addition the T(Tx~x)~Tx is inconsistent with FSL, since it implies

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[Ch. 13

w67. Fitch's models Despite the negative result of w we can adapt the supervaluation semantics of w to validate the assumption that T recognizes its consistency and its closure under logical consequences, plus the adequacy schema with respect to T-free atomic sentences. The result is implicit in Fitch (1963), and so we speak of Fitch's models. We prove that V F - c a n be consistently enlarged by accepting, as new schemata, T ( T ~ A ~ - ~ T A ) , T ( T ( A ~ B ) ~ ( T A ~ T B ) ) , and closure under a stronger T-introduction inference. In particular, we can infer TA, whenever the formula obtained from A by replacing T with the necessity operator Vl, is derivable in a quantified extension of deontic logic (see Bull-Segerberg 1983). Technically, we first refine the basic relation ]]-o of w 67.1. DEFINITION. Fix JI~I=OP-; recall that an e-atom has the form t - s, N t or the negation thereof. (i) Diag(.At~)"- {J~([A])" Jlt~I- A, A e-atom of s (ii) Recall that S E N T ( J I I , ) - {.AI,([A])" A s

X C_ SENT(Jfb) is Jfb-normal iff the following closure conditions are met" 1) X contains Diag(Jl~); 2) X E CONS(JI~); 3) X is closed under logical consequence: 3.1) if A is a logical truth in the language L(.A~), then .A~([A]) E X; 3.2) X is closed under modus ponens: a E X and (a---~b) E X imply bEX. (iii)

.AI, N O R "- { X C M" X is ~t-normal}; .)~-NOR(Y) "- { X C_ M" Y C_ X and X E .Ate-NOR}.

If Y E .Ate-NOR(X), we say that Y is a normal extension of X. (iv) Once ~ is fixed, recall that

J ( X ) "- {MI~([A])" A is an L(Ml~)-sentence with (~t~,X)I-A}. We observe a number of useful facts on 3b-normal sets. The verification is an easy exercise. 67.2. LEMMA (i)

Existence of Jft,-normal sets: if X C_ M, then J ( X ) is J~-normal.

(ii)

If Y C_ Jfl,-NOR and Y is non-empty, then M Y is Jl~-normal.

XIII.67]

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Pitch's Models

(iii) If Y C MI~-NOR and Y is C -directed (i.e. for every X, Y E Y, there is some Z E Y with X C_ Z and Y C_ Z), then U Y is .A~-normal. 67.3. DEFINITION. Let X C_ M, A s (i) XI]- 2 A iff for every Y E .Ag-NOR(X), then Y]-A; (ii) (I)2(X) "- {~([A])" XII-2A}; FIX2(atg ) :- { X C_ M" X - (I)2(X)}. Clearly (I)2 is monotone and hence FIX2(.Ag ) is non-empty. (iii) We set Nc~(a?l~) "- the C_-least element of FIX2(.AI, ). As usual X C_ M is (I)2-dense iff X C_ (I)2(X). 67.4. LEMMA (i) If tf is a C -directed family of .A~-normal r then U Y is Jtl~-normal and r

subsets of M,

(ii) if X is .~-normal, ff2(X) is J~-normal; (iii) ff2(0) is Jll~-normal. PROOF. (i): immediate from 67.2 (iii) and ff2-monotonicity. (ii): consider the family Y ( X ) - {J(Y)" Y E Jtl~-NOR(X)}. Then Y(X) is a non-empty family of Jll~-normal subsets by 67.2(i). But f f 2 ( X ) - A Y(X) and the conclusion follows by 67.2 (ii). (iii): (I)2(q)) - A 3'(0) and Y(q)) is a non-empty family of R - n o r m a l subsets (apply 67.2 (i)-(ii)). E! 67.5. LEMMA. Let X E Jtl~-NOR; then (i) XII-2A implies XI= A (A arbitrary sentence of L(MI~)); (ii) XI= T ( A ~ B) ~ ( T A ---, TB); (iii) if A is a sentence of Lop(.A~ ) of the form t - s, ~ t -

s, Nt, -~Nt,

XI---A~TA;

(iv) if X is r

, XI= T A ~ A;

(v) if X E FIX2(alI~ ), XI= V x T A ---+T V x A . PROOF. Assume that X is alg-normal. (i)-(ii) are trivial by definition and assumption, while (iv)is a consequence of (i) with (I)2-density. (iii). Let A = -~(a = b). If XI=-,(a = b), then .Agl=-~(a = b) and hence Jll~([--,(a = b)]) E Diag(Jfg) C_ X by normality of X, i.e. XI=T--,(a = b). Conversely, if XI=T--,(a = b) and

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[Ch. 13

Xl= (a - b), .A~([a - b]) E Diag(J~) C_X, against the consistency of X. The other atomic cases are similar. (v). Assume XI=VxTA; by (I)2-density , XII-2A(a), for each a E M, whence XII-2VxA, which finally implies, by (I)2-closure , XI= TVxA. [:] 67.6. LEMMA. Ncc(atg ) is ~-normal. Hence FIX2(atg ) n atg-NOR is non-

empty. PROOF. Observe that N~(Mg) - U {N(c~)" c~ E ON and c~ > 0}, where N(0)-0, N(/~+l)-O2(g(/~))and N(A)-U{N(6)'6 card(M). 68.3. DEFINITION

In(X) C_X(7)

and

(ii) A limit ordinal 6 stabilizes (A,X) iff 6 covers (A,X), I n ( X ) and Out(X) - Out(X, 6).

X(6)

(i) A limit ordinal 6 covers Out(X) M X(7 ) - 0 , for every "/> 6.

Clearly, if 6 covers (A,X) moreover, Out(X) C_Out(X, 6).

and

(A,X)

6 a with ti < R(M), which

covers (A, X). PROOF. If a E In(X) (a E Out(X)), we set:

Height(a) "- the least c~ such that a E X(fl)(a ~ X(fl)), for all/3 _> c~. Then card({Height(a)'a C Stab(X)}) c~ and 6 > sup(Height(a)" a E Stab(X)). By choice, 6 covers (A, X). [3 68.5. T H E O R E M (Stabilization). Let (A,X) be a process on M. Then for every a < R(M), there exists an ordinal 6 with a < ti < R(M), which stabilizes (A, X). PROOF. We show that it is possible to filter out all the unstable elements which possibly enter X(6) (6 being an ordinal given by covering). To this aim, we choose a limit ~ < R(M) and an enumeration {a(fl): fl < )~) of Unstab(X), where each element occurs infinitely often (for every a = a(~) with ~ < ~, there always exists u < )~ with a(~)= a(u) and ~ < u). We recursively define a strictly increasing ordinal sequence {f(~): ~ < A} of length A, whose terms are < R(M):

f ( O ) - min{7"~ < 7 and 7 covers (A,X)};

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[Ch. 13

f ( # + l ) -- the least 7 > f ( # ) such that a(#) E X(7) (a(#) ~ X(7)) if a(#) it X ( f ( # ) ) (if a(#) E X ( f ( # ) ) , respectively);

f(~) - least 7 > sup{f(fl)" fl < ~}, if ~ is a limit. The sequence is well-defined by covering lemma and the choice of the enumeration of unstable elements; moreover, by definition, if fl < ~ < A, f(fl) < f(~). Hence 5 - sup{f(~)" ~ < A} is a limit < R(M) and it trivially covers (A,X). It is enough to check that for every a E X(5), a ~ Vnstab(X). By contradiction, assume a E Unstab(X) and a E f3 {X(fl)" a < fl < 5}, for some ~r < 5. Since f is increasing, there is some ~ < A such that a < f(~) and hence: Vfl(f(~) _< fl < 5----~a E X(fl)).

(1)

Using the enumeration of unstable elements with infinitely many occurrences of each term, we must have a - a(~) for some ~ with ~ < 71 < A, whence ~r < f(~) < f(~) < 5. But (1) implies a E X(f(~7)) and by construction of f, a ~ X ( f ( ~ + l ) ) , against a E M {X(fl)" ~r _ fl < 6}. U! 68.6. DEFINITION. If (A,X) is a process on M,

(r(A,X) "-- the least stabilizing ordinal or the closure ordinal of ( A , X ) . 68.7. LEMMA (i) ( X ( a ) ) ( f l ) - X(a+fl); (ii) If X ( a ) -

X(fl), then X ( a + 7 ) -

X(fl+7).

PROOF. (i): induction on ft. (ii)" immediate by (i). [1 68.8. T H E O R E M (Periodicity). Let a be the closure ordinal of ( A , X ) on M. Then there exists exactly one ordinal r - r X) < R(M), the so-called "period", such that: (i) X ( ~ ) (ii)

X(~+r

for every ordinal 6;

if a < a, there is an ordinal v < r with X ( c ~ ) - X ( a + v ) .

PROOF. Let r be the least ordinal > ~, which stabilizes (A,X) and let r We check (i) by induction on 6. Let 5 - f l + l : then, since X ( ~ ) - X ( ~ ) - I n ( X ) , we have by 68.7 (i) and IH" =

Let 5 - A be a limit and assume that for all v < A, X ( ~ ) - X ( c r + r X(~r) C_ X ( a + r holds, since r stabilizes (A,X).

XIII.68]

Introducing Semi-Inductive Definitions

393

Conversely, if a E X(a+r there is some ~ < or+CA, such that a E X(~) for every /3 satisfying ~ ~r+r such that Y = X(a). By periodicity, Y = X(c~)= X(cr+T/), for some 7/< r Hence Y E Cycle(X). (ii): if X ( a ) E Init(X), a ~- or; were a < a, for some u < r X ( a ) = X ( a + u ) (by periodicity) and X(c~) E Cycle(X); (i)implies a contradiction. (iii): from ( i ) a n d (ii). l-1 To sum up, the behaviour of I m ( A , X ) is already determined below the ordinal ~+r I m ( A , X ) splits into: 1) an initial piece below a; 2) a cycle consisting exactly of the cofinal sets of Conf(X), I n ( X ) being among them. Clearly, if A is monotone and X is A-dense (i.e. X C_ A(X)), the period is zero and the cycle is empty. As to the elements of M, which are unstable with respect to (A, X), they can be characterized as follows:

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[Ch.13

68.12. THEOREM (Characterization) (i)

M C y c l e ( X ) - I n ( X ) - X(cr);

(ii)

U Cycle(X) - In(X) U Unstab(X);

(iii) Out(X) - M - U Cycle(X) and Unstab(X) - U Cycle(X)-X(a). PROOF. (i) I n ( X ) - X(r MCycle(X)is an immediate consequence of stabilization theorem and 68.11 (i). The periodicity theorem guarantees that if a ~ X(~), there is fl such that ~r _< fl < ~ + r with a ~ X(fl); hence a ~ f3 Cycle(X) (for arbitrary a E M) and f3 Cycle(X) C X(~r). (ii). From left to right, the inclusion is immediate by the properties of ~r. If a ~ Vnstab(X) and a ~ In(X), then a E Out(X), whence there is some 7 > ~ such that a ~ X(fl) for all fl > 7- By periodicity, X(7) - X(~+~/), for some ~/< r and hence a ~ X(~r+r/), i.e. a ~ U Cycle(X). Conversely, let a ~ U Cycle(X), then a ~ In(X); were a E Unstab(X), then a E X(~r+r/), for some ~/< r (r period, apply 68.8). Hence a E U Cycle(X)" contradiction! (ii) immediately yields (iii). I"1

w69. Semi-inductive models for reflective truth. The application of semi-inductive definitions to the semantics of selfreferential systems is due independently to Herzberger (1982) and Gupta (1982). Applications to the modelling of axiomatic systems for truth and property theory can be found in Turner (1987) and Friedman-Sheard (1987). Turner (1990) observes that the internal logic of truth, which is sound with respect to semi-inductive interpretations, is rather rich. However, in view of the inconsistency theorem of w the logic of stable truth cannot in general contain the schema T+-univ, as claimed by Turner (1990) (1990a). As far as we know, there is at present no completeness result, which fully characterizes the logic of truth revision (possibly involving some form of infinitary logic). The aim of this section is quite modest" we apply the new tools of w68 to make clear that there are a few principles, which separate the logics of truth (sound for the supervaluation models and the provability interpretation of Ch. XII), from the logics of truth based on semi-inductive models. 69.1. DEFINITION (i) As in w if .AtI-OP-, M is the domain of ~ , X C_ M, A is a sentence of s X ] - A means "A holds true in the structure (MI~,X/" , i.e. A is true, whenever s receives its usual interpretation in .A~ and

Semi-Inductive Models for Truth

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(dill, X)I= Tt iff 31~(t) E X (36(t) being the value of t i n 31~). (ii) If 31~ is fixed and X C_ M,

J(X) "- {dtl~([A])" A sentence of L(MI,)such that XI= A). 69.1.1. CONVENTION. We restrict our investigation to processes of the form (g,x), where J" @ ( M ) ~ ( M ) i s defined as above, Mt~i=OP-, X C M. The notions of stable, unstable, stabilization ordinal, etc. are referred to the process (J,X). Typically, In(X) represents the set of those sentences of s which are stably true, insofar as we choose X as initial value for the truth predicate T. For simplicity, we identify the elements of In(X) with the corresponding sentences; the elements of In(X) are simply called X-stably true sentences. We shall proceed to check that I n ( X ) i s a model of V F - (see w plus suitable additional T-schemata. 69.2. LEMMA (i) Soundness of stable truth: for every X C M and every

L(.~)-sentence, I n ( X ) ~ TA --, A. (ii)

Consistency: for every X C M, c~ > O, X(a) is consistent.

(iii) I n ( X ) l - T-~A implies A E Out(X). (iv) In(X)I- T-~TA ~ T~A. (v) O u t ( X ) - {MI~([-~A])" In(X)I--T-~A). (vi) S t a b ( X ) - {~([A])" A L(Jft~)-sentence with In(X)I= T A V T~A). PROOF. (i) If cr is the closure ordinal of (J,X) on M, I n ( X ) - X(a); so X(~r)I-TA implies that A is X-stably true, whence .Ate([A])E X(cr+l), i.e. x ( ~ ) l - A. (ii): immediate by induction on a. (iii) If X ( ~ r ) - In(X)l= T-~A, then ~t~([-~A]) E X(6), for every 6 _> ~r. Hence by (ii) ~ ( [ A ] ) ~ X(ti), for every 6 >_ cr, i.e. by definition ~t~([A]) E Out(X). (iv) By assumption, there exists some 6 such that for every f l > 6, ~([-~TA]) E X(fl). We inductively verify that .AI~([-~A])E X ( f l ) f o r all >_ 6+1. Let fl - 7+1 >_ 6+1. By assumption Jtl~([-~TA]) E X(fl+l), whence by definition of the process, X(~)I=-~TA and X(7)I=~A, which implies ~ ( [ - A ] ) E X(fl). If fl is a limit > 6+1, we have by IH dtl~([-A])E X ( 7 ) f o r every 7 with 6+1 (r§ Indeed, if J~([A])EOut(X), then, by definition of stabilization ordinal 3t~([A]) ~ X(a) for every a _ a (here X ( a ) - In(X)) and hence X(a)I=-~TA, i.e. dlt([-,TA])EX(a+I). If a--)~ is a limit >_ (r-i-l, by IH dtl~([--~TA])E X(~) for every 13, q + l _ ~r+l, whence it will follow by choice of ~r, X(~)l= T'dxA. The case of fl limit is trivial by IH. Let fl - 7+1 > ~r+l. By assumption, for all a E M, A(a)E X(a); as ~r is the stabilization ordinal, A(a) E X ( 7 + l ) for every a E M, whence X(7)I= VxA, i.e. VxA E X ( 7 + l ) . M-rule.

T-intro: it suffices to check X(~r)I=TA--.TTA. Assume X(~r)I=TA; we inductively prove that T A E X(fl), for every fl > cr+l. The limit case is trivial by IH. If f l - ~+1, observe that the assumption A E X(cr) implies A E X(8), if ~r < ~, i.e. X(~)I= TA, whence T A E X(8+1). T-elim: by 69.2 (i) X(~r)l= T T a ~ Ta. ~T-intro: apply T-intro, closure of X(~r) under modus ponens and the fact that T-cons E X(cr). -~T-elim" if X(~r)I=T-~TB, we inductively verify Vfl > c~+l.(-~B)E X(fl), where a is an ordinal such that Vfl > a.(-~TB)E X(fl). The limit case is trivial by In. If f l - 6+1 > a + l , ( - T B ) E X(8+2), hence X(6+I)I=~TB , i.e. X(6)I=-~B, and finally (-~B)E X(6+I). As a consequence of claims (1)-(2), we have that LIS- C_ILST. E! 69.5. COROLLARY. Let ~1~ be an w-standard model of O P - and let X C M. Then In(X)I=LIS, but I n ( X ) I = - - , T ( V x T A ~ T V x A ) , for some sentence A. PROOF. The first part is obvious by 69.4 and assumption. As to the negative claim, observe that if I n ( X ) I = T ( V z T A ~ T V z A ) , then I n ( X ) would be ;v-inconsistent by 65.4. But I n ( X ) is w-consistent, by closure under M-rule and the fact that the extension of the predicate N is a class ! D We now turn to the question of characterizing the external logic of stable truth, namely the set E L S T - - { A - A L-sentence such that In(X)l= A, for every .AI~I=O P - and

XC_M}. A natural (finitary) approximation to ELST is suggested by the previous results.

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69.6. DEFINITION (i) Let T+-Rcomp (T+-S4comp, T+-S5comp) be the schema which is obtained by prefixing T to the schema T-Rcomp (T-S4comp, T-S5comp respectively) of 69.3. For instance, T+-Rcomp has the form:

T[T(TA ---,A)--, (TA V T-,A)]. (ii) LES- is the theory which contains classical predicate logic with - , the non-logical axioms of OP-, T-out, T+-elem, T+-cons, T+-imp, T-rep, T-univ, T-negT, T+-log, T+-Rcomp, T+-S4comp, T+-S5comp (see 65.1, 59.1, 69.3). (iii) LES "- LES-+ the internal N-induction schema

T[(A(0) A V x ( A ( x ) ~ A(x+I)) ~ Vx(Nx ~ A(x))]. Notice that LES extends the system FT of 67.7. By a straightforward argument, we obtain: 69.7. LEMMA. LES- (LES) is closed under the rule: if L I S - F A, then L E S - F TA (LES F TA).

~l= OP-, x c_ M, then In(X)l= LES-. In addition, In(X)]= LES, if ~1~ is w-standard.

69.8. THEOREM. If

PROOF. All the relevant work has been done above: we simply apply 69.2 and 69.4. D 69.9. REMARK. F T - is strictly contained in LES- by 67.12. In particular, if S is a term such that S - ITS], then L E S - F TS V T-~S, while we know that TS V T-~S is unprovable in FT-. 69.10. Problems. Is T+-S5comp independent from FT-? What is the proof-theoretic strength of FT-, LES- ? We conjecture that they are equivalent to VF-.

CHAPTER 14

E P I L O G U E : A P P L I C A T I O N S AND PERSPECTIVES w w w w

~i74. w

A logical theory of constructions: informal motivations A logical theory of constructions: basic syntax Axioms for the computation relations Extending the logical theory of constructions with higher reflection Proof-theoretic reduction Perspectives: related work in Artificial Intelligence and Theoretical Linguistics Sense and denotation as algorithm and value: subsuming theories of reflective truth under abstract recursion theory

Confronting a theoretical piece of work with applications is always useful for a critical assessment. For this reason, we address the question of relating the systems of reflective truth that we have been investigating so far, with applications in Theoretical Computer Science (TCS), Artificial Intelligence (AI), Linguistics. We are concerned only with potential connections, and not with direct, well-established applicatzons, already available in the literature. We shall consider three examples: a) a logical theory of constructions, arising from TCS, and its modeling in the systems of chapters X-XI; b) some logics, motivated by knowledge representation and the semantics of natural languages; c) Moschovakis's intensional approach to the foundation of the theory of algorithms. We underline that our choice is largely a matter of taste and strongly bound to the limited competence of the writer. Thereby, the aim of the present chapter is rather that of putting the content of the book in a wider perspective and suggesting new problems; there will be no attempt of systematization, nor we try to supply complete details. The only relative exception is the first example, dealing with the logical theory of constructions, LTCw; but this is due to the fact that LTCw fits nicely with the material of chapters VIII-XI. As to the examples of part b), we hope that some applicative-minded reader will find the results of chapters XIIXIII, as well as those of chapter VI, of some interest. The final example, which discusses Moschovakis's lower predicate calculus with reflection, is to us highly suggestive: it should lead to reflections, embracing both the foundations of recursion theory and formal semantics.

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w70A. A logical theory of constructions: informal motivations In TCS new logical formalisms are currently investigated: (i) as tools for representing, stating and establishing properties of programs (e.g. equivalence, termination and correctness); (ii) as tools for program extraction; (iii) as tools for reasoning about the specification of programs and their typing (foundations of type theories); (iv) as abstract theories of computation over abstract data types. This list is not exhaustive and the single aims (i)-(iv) are usually integrated, being a distinctive feature of the logic methodology its unifying power. In this respect, we may mention Martin-LSf's type theories (Martin-LSf 1984, B.NordstrSm et al. 1990), the ELF-approach (Harper-Honsell-Plotkin 1987), the theories of constructions (Coquand 1985), NUPRL (Constable et al. 1986), the logical theories of constructions of Aczel et al. (1991), Feferman's theories and its outcomes (Feferman 1979, Hayashi-Nakano 1988, Feferman 1990, 1991a, 1992, Talcott 1992), the proofs-as-programs approach, as developed by Schwichtenberg (1991). We concentrate upon a single example, which appears close to the spirit of this work: the logical theory of constructions LTC, as it is outlined in Aczel-Carlisle-Mendler (1991). On the conceptual side, LTC-theories are motivated by "the idea that the notions of proposition and truth are, after all, the fundamental ones for logic and that the logical notions are the fundamental ones for a deductive system for mathematics. According to this idea, although the notion of type is also essential for mathematics and computer science, it is less fundamental conceptually" (see p. 5, cit.). Technically, we can summarize the basic features of LTC in the following points: 1) LTC includes the values of a functional programming language, as well as the propositions of a reflective logic; in particular, in LTC there is a truth predicate, which expresses the fact that a proposition, as an object of our universe, is true; since the underlying logic is constructive, LTC-higher systems are endowed with predicates expressing the fact that certain objects are propositions (of a given level); 2) the functional language is untyped; but the semantics is operational, explicitly controlled by a "lazy evaluation" relation; 3) the basic equality relation should be decidable; however, in the strongest theory of Aczel et al., there is a mixed approach: recursive aspects

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are handled as equalities, while discrimination and selection aspects are maintained at an operational level; we do not know whether the resulting conversion relation is decidable. An interesting feature of LTC is that it points to possible refinements of the underlying combinatory logic of OP, which make sense of important distinctions for applications, apparently inaccessible within M F - and its extensions. In fact, a limit of our systems concerns equality: we only deal with a single basic equality = , which is interpreted as equality in combinatory algebras; hence - is generally undecidable. Moreover, one would like to have a notion of "value" and hence a predicate of definedness, in order to explicitly control the main properties of programs. As we shall see, the system LTC 0 of the next section offers a viable alternative, by introducing a different semantics underlying the theory of programs. A final point of interest is that LTC-theories establish a sort of natural bridge between Martin-Lbf's type theories and the predicative systems of reflective truth with variable levels of part D.

w70B. A logical theory of constructions: basic syntax In this section we are going to introduce expressions with arities and their basic definitional equality - , together with the notion of canonical realization (term models) for the resulting formalism. 70.1. (i) Arities: they are inductively generated by the following clauses: OB (individuals), BOOL (formulas) are basic arities; if c~, /3 are arities, so By currying we also write (ch...c~n)---,c~ for (ch~(c~ 2 . . . ( % ( - o a ) . . . ) . Every symbol is assigned an arity. If we understand BOOL as the arity of formulas, (OB~BOOL), (OB, OB)~BOOL will obviously represent the arities of unary and binary predicates (in the given order); on the other hand (OB, OB)~OB is the arity assigned to binary function symbols. It is clear that the stock of basic arities can be conveniently expanded, insofar as we need new basic sorts of entities. (ii) The formal language s is given by specifying a list of primitive symbols, together with their corresponding arities.

Individual symbols (arity OB): a denumerable list of individual variables (x,y,z,u syntactical variables); the constants 0 and J_ (the object representing the absurd proposition 1 );

Propositional symbols (arity POOL): I (absurd proposition);

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Logical symbols: ---, (implication of arity (BOOL, BOOL)-,BOOL); Vo (universal individual quantification of arity (OB---,BOOL)---,BOOL); V1 (universal function quantifier of arity ((OB~OB)~BOOL)---,BOOL); Predicate symbols: arity (OB, OB)~BOOL: = (equality), LEV (lazy evaluation), N E Y (full evaluation to numbers); arity (OB---,BOOL): T (truth); Pi (proposition of level i), for any i > 0.

Function symbols: arity (OB---,OB): a denumerable list of unary function variables f (f, g, h syntactical variables); S (successor); Inl, Inr (projections); V0; Vl (internal quantifications); P i , for each i > 0; T; arity (OB, OB)---,OB: Pair; -=, ; LEV, NEV, - ; arity (OB~OB)---,OB: A (abstraction); arity (OB, (OB, OB)---,OB)~OB: Spread; arity (OB, OB---,OB, OB--,OB)~OB: Decide; arity (OB, OB, OB--,OB)---,OB: Decidenat; arity (OB, OB, OB---,OB)--,OB: Ind (primitive recursion operator); arity (OB, (OB---,OB)~OB)---,OB : Pa (permuted application). 70.2. Expressions of "~TC: they are inductively generated from the set of basic symbols by means of the inductive clauses for abstraction and application: (i) if E is an expression of arity a and x is a variable of arity fl (hence

t3 = OB or ~ = OB~OB), then (x)E is an expression of arity (fl---,a); (ii) if E is an expression of arity a---,~ and E' is an expression of arity a, then E(E') is an expression of arity/3. 70.3. Notations. Expressions of arity BOOL are identified with formulas and A, B, C play the role of metavariables for them; VxA := V0((x)A ) and Vlf. A := V((f)A). Expressions of arity OB are the usual individual terms; we let Ax.t stand for A((x)t). Multiple abstraction is reduced to iterated abstraction (currying)in the usual fashion; for instance ( x y ) f ( x , y ) i s an abbreviation for (x)((y)f(x, y)). 70.4. Dotted symbols: they are function symbols that allow to associate to each expression of arity POOL an expression of arity OB. In particular, we have:

70.4.1. FACT. To each formula A of s we can effectively associate a term Jl such that )1 has exactly the same free variables of A. The intuitive meaning of the basic function symbols can be clarified by anticipating that the following defining equations are valid in the standard denotational semantics:

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70.5 (Re):

Pa()~(f),h)- h(f);

(Decidenat):

Decidenat(O, x, f) - x and Decidenat(S(y), x, f) - f ( y ) ;

(Spread):

Spread(Pair(xl, x2),h ) - h(Xl, X2);

(Decide):

Decide(inl(x), f, g) - f(x); Decide(inr(x), f, g) - g(x);

(Ind):

Ind(x, y, h) - necidenat(x, y, (x)h(x, Ind(x, y, h))).

It is understood that the terms involved have the appropriate arity. If we define

Ap(x, y) := Pa(x, (f)f(y)), then (Re)implies (/3)-conversion. Ap(~(f), y ) = f(y). In the present proposal denotational equality is split into finer relations, which also take care of the operational level. More precisely, while the Ind-equation is integrated in a suitable definitional equality on expressions, the remaining equations are transformed into inductive clauses, which define an appropriate evaluation relation. The first step takes inspiration from Martin-Lhf's theory of expressions. 70.6. D E F I N I T I O N (i) If ~ is an arity and E, E ' are expression of arity ~, we inductively define the (ternary) relation E - E':~, to be read as E and E' are equal expressions of arity ~. We write E: cr as an abbreviation of E - E: v~; E: cr means that E is of arity ~. E - E': cr is the smallest relation, which meets the following conditions: 1. Reflexivity: E -

E: ~;

2. S y m m e t r y : E -

E " ~ implies E ' -

3. Transitivity: E - E': a and E ' -

E: or; E": cr imply E - E": a;

4. c~-conversion: if x and y are variables of arity cr and y does not occur in E, then (x)E - ( y ) E [ x :-- y ] : a; 5. /?-conversion: if E': a, x: a, E: 5, then ((x)E)(E') =_E[x := E'] : ~; 6. ~-conversion: if E:cr-.5, x:cr, then ( x ) E ( x ) - E:~-~5; 7. ~-conversion: if x: a and E - E': 5, then (x)E - (x)E': (r--,5; 8. application: if E -- E': a - ~ i and H -- H': or, then E(H) - E ' ( H ' ) : li;

9. definiendum- definiens: E -

E': OB, provided E, E ' are the expressions involved in the (Ind)-equation of 70.5 above.

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(ii) Let - ~ := { ( E , E ' ) " E -

E':cr}; [E]~ "- { E " E -

~E'};

M~ "- {[E]~: E closed expression}. If c r - - ~+7, [E]~ E M~, and [F]~ E M~, we put [F]a([E]~)- [F(E)]~. According to the last definition, each element of M ~ + ~ represents a unique function from M~ into M~. Henceforth we use the families M ~ , - ~, where a is an arity, in order to define a Tarskian semantics for formulas of LTC o. 70.7. DEFINITION. (i) A canonical . A "- (M,s,Mo.,~), such that" 1. ~ . - O B ,

realization

for s

is a triple

o-.-OB--.OB;

2. r is an interpretation function satisfying the conditions: 2.1. for each expression E of arity a, r

[E]~;

2.2. ~( - ) - the relation - ~;

2.3 for each i E ~o, r 2.4. r

r

and r

are subsets of M0;

are subsets of M~ x M~.

(ii) The relation v~l=A (A sentence of LTC0) is inductively defined according to the standard Tarskian clauses, once we stipulate that: 1. variables of arity O B range over M6, while variables of arity O B - ~ O B range over M~; 2..At, l: (t - s) iff t - s "~; .Atl= T t iff [t]~ E r ~ l = P i t iff [t]~ E O(Pi) (i E or, ~ -- OB, t, s closed terms, i.e. closed expressions of arity OB).

We conclude with a remark: if we omit the defining equations for Ind, the expressions with the family { - ~ ' a arity} yield a (version of) typed A-calculus. As usual, the definitional equality relations can be generated by the corresponding natural reduction relations, which are introduced by regarding the basic --clauses as contractions. Moreover, every expression reduces to a unique normal form (and actually every reduction sequence is finite; strong normalization). As a corollary, one has: 70.8. PROPOSITION. The relations are decidable.

-~

without the Ind-defining clause

It is not known to us whether the unrestricted relation with Ind-clause E-E':OB is decidable. The problem is left open by Aczel, Carlisle, Mendler (1991); according to them, let =~+ be the reduction, generated by

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(a)-, (fl)-, (~/)-contractions, extended with the (Ind)-contraction:

Ind(t,s,h) :=~+ Decidenat(t,s, (x)Ind(x,s,h)). Then :=:~+ is still consistent, in the sense that a corresponding ChurchRosser property holds (if E:=~+E ' and E=:~+E '', then E'=:~+H and E"=:~+H, for some expression H). As a consequence" 70.9. The following basic special equality axioms become true in the corresponding term model:

F ( f l , ' " , f n) -- F ( g l , ' " , gn): OB =~f i(ul,..., urn) - gi(ul,..., urn): OB; -~F(ci. . .cn) - G(hi. . .hn): OB; here F, G are primitive function symbols ~ Ind, and fi, appropriate arities.

gi, uj have

w71. Axioms for the computation relations The basic syntax contains a functional programming notation; so we have to explain how to compute programs and how to identify them. Of course, one could stick to a denotational semantics: the resulting interpretations would be essentially the (many sorted versions of) models of A-calculus, which were introduced in Aczel (1980) under the name of lambda structures. By contrast, following Aczel-Carlisle-Mendler(1991), we outline a semantics which lies between denotational and operational semantics. First of all, we specify the space of values. This is done by exploiting the idea of lazy evaluation and Martin-LSf's distinction between canonical and noncanonical expressions. Roughly speaking, a canonical expression is an expression, which directly manifests the data type it belongs to and can be immediately understood in terms of the givens we are dealing with (numbers, functions, lists, etc...); as such, it is a static object. On the contrary, non-canonical expressions involve control features, that have to be eliminated, in order to understand the direct meaning in terms of the givens. Of course, this is vague, but it will be made precise by specifying canonical forms and by inductively defining the appropriate evaluation relations. 71.1. DEFINITION. (i). Canonical symbols" 0, S, A, Pair, Inl, Inr, -:~, _J_,

~]0, ~/1' --'

L E ? , N E V , Pi (i > 0), T;

(ii) a term t of arity OB is canonical (or is in canonical form) iff either t is 0, or else its outermost function symbol is canonical.

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The computation process is formalized by the predicate L E V of lazy evaluation: L E V ( t , s ) holds if t evaluates to s and s is in canonical form. L E V is inductively defined by appropriate inductive clauses that involve non-canonical symbols. But L E V does not suffice in general: for instance, S(t) is canonical, but we cannot directly read off from S(t) that S(t) truly represents a natural number, unless we already know that t represents a natural number, i.e. either t is 0 or has the form S(r). The second case may require further evaluation and so on. To sum up, we also need a primitive predicate N E V ( t , s ) which holds exactly when t fully evaluates to a numeral and hence we properly have that t represents a natural number. Again, this leads to an inductive specification of N E V , which involves primitive recursion and allows to introduce a natural number predicate. Since the reduction relations associated to L E V and N E V are Church-Rosser, we shall be in the position to define the appropriate equivalence relations on programs and numbers. 71.2. D E F I N I T I O N (Computation triples). We say that (a,b,k) is a computation triple (in short ( a , b , k ) E C O M P T ) iff one of the following (mutually exclusive) cases holds: (i)

a -- )~(f), b -- h ( f ) and k - ( x ) P a ( x , h ) ;

(ii)

a - Pair(xl, x2) , b - h(Xl,X2) and k - (x)Spread(x,h);

(iii)

a-

Inl(u), b - f ( u ) and k - ( x ) n e c i d e ( x , f, g);

(iv)

a-

Inr(u), b - g(u) and k - ( x ) D e c i d e ( x , f, g);

(v)

a - 0 , b - u and k - (x)necidenat(x, u, g);

(vi) a - S(z) , b - g(z) and k - ( x ) n e c i d e n a t ( x , u, g). {For a motivation of 71.2 simply recall the equations of 70.5}. 71.3. The ground system LTC 0 (without reflection) LTC 0 is the theory in the given language LTC above, which contains the following axioms: 71.3.1. two sorted intuitionistic logic with standard equality axioms (for objects); recall that the basic sorts correspond to variables of arity O B (x, y, z...) and to arity of O B - ~ O B (f, g, h,...), respectively. In particular, we have the axiom schema:

V f A ( f ) - - - , A [ f "- g] (g term of arity O B - ~ O B , free for f in A). We also postulate two special principles for - , which are related to the fact that the intended model is a term model (see 70.9):

Axioms for the Computation Relations

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SEI:

F ( f l , " ", f n) - F(gl," " "' gn) ~ f i(ul, "'" , urn) -- g i ( u l , ' ' ' ' urn);

SE2:

-~F(Cl, . . ., ca) - G(hl, . . ., ha);

here F , G are primitive canonical function symbols; fi, gi, uj have appropriate arities (remember that - only applies to expressions of arity

OB). 71.3.2. Closure (OB~OB)~OB:

(~)

under

definitions

for

arities

OB~OB

and

Vw(((v)t)(w) - ((y)t[v . - y])(w)) (provided y does not occur in t); Vf(((g)t)(f)-

(fl)

explicit

((h)t[g "- h])(f)) (provided h does not occur in t);

Vu(((x)t)(u) - t[x : - u]) (provided u is free for x in t); V f ( ( ( g ) t ) ( f ) - t[g "- f]) ( provided f is free for g in t).

71.3.3. LEV-axioms: LEI"

L E V ( c , c), provided c is a canonical term; L E V ( x , y ) ~ L E V ( y , y); (if x is evaluated to y, y is canonical);

LE2"

V x V y ( L E Y ( x , y) A L E V ( x , z) -~ y - z);

LE3"

LEU(x,a)--.(LEW(k(x),z)~

L E V ( b , z ) ) , for (a,b,k) C COMPT.

71.3.4. N E V - a x i o m s : NE1-

L E V ( x , O) --~ N E V ( x , 0);

NE2"

LEV(x,S(y)) A iEV(y,z)-~

NEV(x,S(z)).

N E V - I n d u c t i o n : if A(x, y) is a formula, m

V x ( L E V ( x , O ) ~ A(x, 0)) A V x V y V z ( L E V ( x , S(y)) A A(y, z ) ~ A(x, S(z))).--~ ---, Y u V v ( N E V ( u , v ) ~ A(u, v)). ( N E V - i n d u c t i o n states that N E V is the least predicate closed under the inductive clauses formalized by NE1, NE2). 71.3.5. Peano axioms: wvy(

s(

) -

0 ^

-

9 -

y));

71.3.6. Primitive recursion:

I n d ( x , y, f ) - Decidenat(x, y, (x)h(x, Ind(x, y, h))). 71.4. DEFINITION

i ( x ) "- N E V ( x , x )

( x is a numerical value);

410

Epilogue: Applications and Perspectives

[Ch.14

Nat(x) "-- 3 y N E V ( x , y) (x denotes a natural number); E q g a t ( x , y) "- 3 z ( g E Y ( x , z) A N E Y ( y , z)); xl "- 3y L E V ( x , y) ( - x has a value or x is defined). 71.4.1. REMARK. LTC 0 can easily interpret a logic of partial existence s la Scott and distinguish between quantifying over all the entities of the universe (just the usual Y and 3) and quantification restricted to values" Yx+(...) "- Yx(xl---+...) and 3 x + ( . . . ) " - 3x(xl A...). The intended models of LTC 0 are obtained as special canonical realizations. 71.5. DEFINITION. Let 5 - OB: (i) LEV is the least relation C_ M ~ x M ~ following clauses:

which is closed under the

1. if t is canonical, then ([t]6 , [t]6) E kEY; 2. if (t, s, r) is a computation triple (71.2), ([P]6, [t]6) E ILEY and ([s]5, [q]5) E LEV, then ([r(p)]5, [q]5) E LEY. (ii) NEV is the least relation C_ M 6 x M 6 following clauses:

which is closed under the

([t]6, [016) E LEV implies ([t]6 , [016) E NEV; ([t]6, IS(r)]6) C LEV and ([r]6, [p]6) E NEV imply ([t]6, S[p]6) E NEV. (iii) The interpretation function ~o satisfies: @o(T) - e~o(Pi) - 0, for each i E w; ~ o ( L E V ) - LEV and r

- NEV.

71.6. THEOREM. If .Ago " - ( M 6 , M6__,6,~o) , then alg0]=LTC 0 (i.e..Ag o is a canonical model of LTCo-axioms ). As to the proof, the crux is to extend the Church-Rosser theorem (w to the definitional equality relation, in order to verify the special equality axioms SE1-SE2 (see Aczel-Carlisle-Mendler 1991). The equality axioms, (c~), (fl), the Peano axioms and primitive recursion are immediately verified, as ~0 is a canonical interpretation. By choice of LEV and NEV, L E V - and N E V - a x i o m s are valid in ~0" What can be said about the strength of LTCo? An answer is given by a simple observation:

XIV.72]

Logical Constructions with Higher Reflection

411

71.7. PROPOSITION (i) PA, the first-order system of Peano arithmetic, is interpretable into LTC o. (ii) LTC 0 is interpretable in PA. PROOF (hint). (i): the domain of the interpretation is the defined predicate Nat, which is closed under successor and contains zero; EqNat interprets equality. The Ind-axioms imply that there are functions + and 9 under which Nat is closed; NEV-induction implies Nat-induction. (ii) One has to formalize the term model construction of w with the corresponding Church-Rosser theorem (see appendix to chapter I). This is possible since the inductive definitions of definitional equality - , and the relations LFV and NFV, being given by existential positive clauses, can be explicitly defined and arithmetized in PA. E! 71.7.1. REMARK. Assume that LTC o F-vx(gat(x)--, Nat(t(x))): then 71.7 implies that the number-theoretic function defined by the term (x)t(x) is provably recursive in PA.

w72. Extending the logical theory of constructions with higher reflection Up to now, the predicate symbols Pi and T have been left undetermined. We wish to add axioms interpreting Pit as "t is a proposition of level i" and Tt as "t is a true proposition". 72.1. DEFINITION. LTCw is the extension of LTC 0 with the following propositional and truth axioms (for each i with 1 ~ i < w): PTli

If c - [ A ] and A - ( r - s), LEV(r,s), NEV(r,s), L ,

LEV(t,c)--*Pi(t); LEV(t,c)---~(A~Tt); PT2i

(LEV(r, t-~s) A Pi(t) A (T(t)--,Pi(s))) ~ Pi(r); ((LEY(r, t-,s) A Pi(t) A (T(t)--+Pi(s))) ~ (T(r) ~-, (T(t) ~ T(s)));

PT.3i

(LEY(r,~/(f)) A V x P i ( f ( x ) ) ) ~ Pi(r); (LEY(r, ~/(f )) A VxPi( f (x)) ) --, (VxT( f (x)) ~ T(V(f))).

PT.4i, j if 1 _< j < i,

LEV(r, Pj(t))--,(Pi(r ) A ( T ( r ) ~ nj(t))); Pj(t)~Pi(t).

Epilogue: Applications and Perspectives

412

[Ch.14

72.2. DEFINITION. Let 1 < i < k. (i)

Internal truth of level i: Ti(t)'-T(t)APi(t

(ii)

).

LTC k is the subsystem of LTC~, which only contains the predicate

symbols P I , . . . , P k . At first sight, one might conjecture that each fragment LTC k is directly interpretable in the system STLR k with reflective truth predicates up to k (see w one would simply be tempted to identify the truth predicate T with Tk, the truth predicate of level k, and to define the notion of proposition of level i (for 1 _< i __ k) with the classical Pit "- Tit Y Fit. The idea, though roughly sound, is not viable, since P i and T must be well-behaved with respect to L E V and N E V ; but these predicates cannot be simply reduced to usual conversion equality, due to the special equality axioms. Of course, we can define new systems corresponding to TLR, ITnr RS n of Chapters VIII-XI, which are based on LTC 0 instead of OP. Then the proof-theoretic reduction of Chapters X-XI can be adapted to the new systems without any difficulty, in order to show: 72.3. THEOREM. For each k, LTC k is proof-theoretically reducible to T L R (i.e. the formal consistency of LTC k is provable in OF + TI(a), for some a < F 0 and hence in TLR; see Ch. XI) The reader not interested in further dreary details, can directly skip to w74. In the rest of w72 and in w73 we illustrate a different route to the theorem, which consists of building up a model of LTCk, directly in the available systems. Since the verification of 72.3 is rather lengthy, we split it in a few steps; in this section we restrict our attention to the interpretation of LTC 0.

First Step. We simulate the term model of LTC 0 within an untyped model of OP by direct use of the combinatory structure available. Abstractions (x)t, (f)t are represented by h-abstraction; the basic identity of LTC k is sent into combinatory equality" thus the interpretation is certainly unfaithful. Nevertheless, the interpretations of LEV, N E V and the basic function symbols are chosen in such a way that the special equality axioms, together with LEV, NEV-axioms, become true. 72.4. DEFINITION (i) We associate to each individual constant and each primitive function symbol G of LTC k a corresponding combinator G* (of OP).

)~* "--)~f.(-8,f); Pa* "-~x~h.(9,(x, hl); (.j_ )* . - ( 2 0 , 1 / ; Inl* "-- )~x.(]-O,x); Inr* "-- Ax.(-H, x); Pair* "- AxAy.(-~, (x, y));

XIV.72]

Logical Constructions with Higher Reflection

413

Spread* "- )~x)~h.(13, (x, h)); Decide* "- )~x)~f )~g.(14, (x, f , g)); Decidenat* : - )~x)~yAf .(1--5, (x, y, f));

(LEfZ)* "- AxAy.(1---6,(x, y));

( N E f / ) * "-- )~x)~y.>; (Va)* "-- )~x.(1---g,x); S* "- )~x.(1---9,x>; O* "--(20,0);

- * := ID;

(Vo)* "- ALL;

~ * "- )~x$y.IMPxy;

(~Pi)* "- )~x.[P*(x)]; (T)* "- )~x.[T~(x)], where 1 < i < k and P*, T i are defined in 73.1 below; Ind* "- FP(~g~x)~yAh.Decidenat*(x, y, )~u.g(u, y, h))). N.B. It is understood that I M P ' - ~ x J ~ y . N E G ( A N D x ( N E G y ) ) ; NEG, A N D , I D , A L L are the combinators of Ch. II, 7.1. F P is the fixed point combinator of Ch. I,w 2. Notice that * actually depends on k. (ii) We inductively extend * to arbitrary (also n o n - p r i m i t i v e ) f u n c t i o n symbols and terms of L T C k of arity OB: 1. if c is a constant (c)* - c*, as defined in 72.4 (i) ; (xi)* - x2i+l if x is a variable of arity OB; (fi)* - x2i if f is a variable of arity OB--~OB;

2. F ( t l , . . ., tn) * " - F* t l * . . tn* (where F* is defined in 72.4 (i), if F is a primitive function symbol of LTCk; F* is defined according to the preceding clause, if F is a function variable f). 3.

(x)t* "- )~x.(t*) and (f)t* - )~f.(t*).

N.B. The use of variables of odd and even index, in order to interpret variables of arity O B and (OB--.OB) respectively, is only required to avoid undue identification of variables (e.g. we want that the translation (f(x))* has two distinct variables corresponding to f and x). Henceforth we omit explicit mention of indices and we still keep using f, g, h as metavariables for variables in function position. By inspection of the definitions above, we obtain the expected independence properties, which also imply the counterparts of the special axioms SE1-SE2 and the standard Peano Axioms: 72.5. LEMMA. I f C and E are distinct function symbols of LTCk, then OP ~- -~C* - E*. Moreover OP proves: SEI*"

F(fi,"',

fn)* - F ( g l , ' " , gn) --*fi(ul, "'" , urn)* - gi(ul, "'" , urn)*;

SE2*"

-~F(ci...Cn)* - G(hl...hn)*;

here F, G are distinct primitive function symbols ys Ind, and f i, gi, uj have the appropriate arities (remember that - only applies to expressions of arity OB);

Epilogue: Applications and Perspectives

414

suc*:

-

5")

A

wvy(s*

-

s*y

-

[Ch.14

y).

Second step. We introduce predicates L E V ~ and N E V ~ , which work as "interpreters" of LTCk-terms. The notion of canonical object is explicitly definable by means of the following formula of STLR k-

72.6. DEFINITION (using 72.4; k >_ 1) C a n k ( x ) :-- 3 y 3 z ( ( x -- .~*y) V (x -- S ' y ) V (x = -0") V (x = I n l * y ) V V (x - I n r * y ) Y (x - ( i )*) V (x - N E G y ) V

(x - (V0)*Y)V

V (x -- (V1)*Y)V (x - ( - ) * y z ) V (x - ( L E V ) * y z ) V Y (x -- ( N E V ) * y z ) V

(x - A N D y z ) Y

V (x ----[Tl(Y)] ) V . . . V (x ---- [Tk(Y)] )

(x - P a i r * y z ) Y

V

V (X : [Fl(Y)] ) V . . . V (x -'- [Fk(Y)])). 72.6.1. FACT: we can find a formula C P T * ( y , w, z), which translates the condition "(y, w, z) is a computation triple" (in the sense of 71.2 above) into the language of STLRk: C P T * ( y , w, z) := 3 f 3 h ( y = ~ * f A h f = w A z = ~x.Pa*xh) V V 3 u3v3h(y = Pair*uv A w -- huv A z = A x . S p r e a d * x h ) V Y 3u3v3n3g(z=)~xDecidenat*xvg A ((y=O* A w=v) Y ( y = S * n A w = g n ) ) ) V

v 3 3f3g(z=

.D cid * fg n

n

v

n

72.7. LEMMA. STLR k F C P T * ( y , w, z) A C P T * ( y , w', z) ~ w - w'. PROOF. We essentially apply lemma 72.5. Assume the antecedent of 72.7 and also z = ~x.Decide*x f g, y = Inl*u, w = f u, for some u, f, g. Then C P T * ( y , w', z) implies by SEI*: z - ~x.Decide *x f ' g', y - Inl(u'), w'-f'u', for some f', g', u'. Therefore D e c i d e * x f g = D e c i d e * x f ' g ' and Inl*u-Inl*u', which yield by SE2* f - - f ' , g=g', u=u', whence f u = f'u', i.e. w = w'. The other cases are similar. 0 Let B k ( x , y , v) be the formula: 3 z 3 a 3 c 3 w ( C P T * ( c , w, z) A x = z(a) A (a, C)~ l v A (w, Y)~71v)

and define L Y k ( X , y, v) := ( C a n k ( x ) A x = y) V Bk(X , y, v). By inspection, we see that ~ v . { ( x , y ) : L Y k ( X , y , v)} is an existential operator (see Ch. II, 10.9) and actually that L V k ( x , y, v) is elementary in v. By 10.9.1 we find a term A u . L E V ~ ( u ) such that:

Logical Constructions with Higher Reflection

XIV.72]

415

LEV~(O)- 0 and L E V ~ ( m + I ) - {(x,y)" LVk(X,y, LEV~(m))}; finally we put LEV~ "- {(x,y)" 3m. (x,y}TI1LEY~(m)}. Now we claim" 72.8. LEMMA. Let k > 1. D

(i) STLR k F Cll(LEV~); (ii) STLR k F (x,y)TIILEV~ ~ LVk(X,y, LEV~); (iii) STLR k F (Cank(x)---~(x,x)~llLEY~)A ((x,y)TI1LEY ~---*Cank(y)); (iv) (v)

STLR k F VxVy((x,y)~llLEY~ A (x,z)qlLEY~---*y - z); STLR k F CPT*(y, w, g) A (x, y)~IILEV~---+

((gx, z)~I1LEV~ ~ (w, z>r/1LEY~). PROOF. As to (i)-(ii), STLR k has A0-N-induction, i.e. number-theoretic induction for formulas built up from atoms of the form Tit, Fit (1 < i < k), t - s , Nt, by means of standard logical operations. Now Cll(LEV~(y)) is A o and we can verify with elementary comprehension for classes of level 1 and A0-N-induction that for every m, LEV~(m) is a class of level 1. Then we apply 10.8-10.9.1. (iii): the first conjunct is a corollary of (ii), while the second is obtained by checking VxVy(Ix, y)711LEV~(m)~Cank(y)) by Ao-N-induction on m. (iv): Ao-N-induction and lemma 72.7. (v): apply (ii), (iii). O 72.9. DEFINITION. NVk(Z , y, v) is the formula: (y=0* A (x,O*)~71LEYk) V 3z3w(y=S*w A (x,S z>~llLEVkA (z,w)YlV). Clearly )~v{(x,y)" Nvk(x, y, v)} is an existential operator; hence elementary comprehension and 72.8 (i) imply, for k > 1" STLR k F Cll(V)----~Cll({(x,y > 9NVk(X,y,v)}. If we recursively define

NEV~(O)- 0 and NEV*k(m+I ) - {(x,y)" Nvk(x,y, NEV~(m)) }, we

can

again

apply

A0-induction

on

N,

in

order

to

check

Cll(NEV~(m)) , for every m. If we argue as in 72.8 above, we conclude:

that

Epilogue: Applications and Perspectives

416

[Ch.14

72.10. LEMMA. Let k > 1. (i)

STLR k F- Cll(NEV~) ,

where N E V ~ "- {(x,y)" 3m((x,y)r/aNEV~(m))}; (ii)

STLR k F- (x, y)rllNEY~ ~ NVk(X , y, NEV~);

(iii)

STLR k ~ (x,-O*)rllLEY~ --, (x,-O*)rllgEY~;

(iv)

STLR k I- (x,S*Y)rllLEV~ A (y,z)rllgEY~ ~ ( x , S * z ) r l l g E Y ~.

72.10.1. CONVENTION: henceforth we adopt the abbreviations

LEV~(x, y) "- (x, y)~ILEV~;

g E Y ~ ( x , y) "- (x, y)rllgEY~.

72.11. LEMMA (gEY~-induction). If k > 1 and A(x,y) is a Ao-formula of STLR k (i.e. built up by means of A, ~, Y from atoms of the form Nt, t - s, Tit , Fit , where 1 __1)"

1.

(a, u)~iV(i ) ~ A(a, u, V(i));

2.

if t is a term of the form ( - *xy), ( ( L E ? ) * x y ) , ( i E ? ) * x y ) , [T~x], [F~x], for 1 0-closed 357, 358 club, s e e closed unbounded set CODE 34O coherent element 204 poset 182 coinduction 177, 178, 207 combinator 14, 16, 20, 21, 27 K, S 14, 15

Index paradoxical 17

443 Crossley 259

combinatorial operation 60, 77

Curry 2, 17, 21, 25, 43, 47, 53, 151 currying 403

combinatory algebra 2, 3, 5, 6, 13, 16

cut elimination theorem for OP cr 327

combinatory logic 13, 14, 15, 16, 18, 21, 25, 34 complement 59

partial 324

complete

primitive recursive 347

for the calculus ~ 364

coherent poset (ccpo) 182

Schfitte's 319

lattice 183, 185, 186

Tait's 320

set 47, 208, 357

weak 295,301

completeness

cut rule 155, 291,315

~op-Completeness 9 363 w-completeness 376

Dalen, D. van 27

complexity

decomposition theorem 393

logical 10, 257, 264-68, 281 285, 293, 295,313, 327, 336, 370

abstract 153

n-complexity 303, 305,307, 324

definition by cases 14, 15

analytical 203

denotation 422

elementary 58, 60, 61, 62,

67,

223, 229, 242, 243, 253 second order 63, 178, 200, 239 type-free 44 computation relations 407 computation triples 408 concatenation 244, 272,341 connection axioms 220 conservation, s e e theorem consistency axiom,

see

axiom

consistency lemma 322, 363, consistency theorem 156 47,

29, 30, 31, 36, 103,

109, 112, 118-120, 144 continuous operator,

see

set

dependent choice,

see

schema

derivability relation for I T ~ 3O5 OP cr 327 RSn316 ]Eo o 158 STLR 291

STLR n 297

constructive completeness 115 27,

dense,

STLR c~ 294

179

Constable 402 continuity

of level n, 157 definability theory 85

comprehension

consistent set

Dedekind 152, 243 deducibility

see

operator

conversion 16

Dershowitz 283 Devlin 232 diagonalization 76, 77 domain operation 59

covering 147 creative set (property)

derivative 270

71-72

dual extensional membership 132, 133

444

Index

effective inseparabilty 108 operator 118 elementary atom 46

extensional 2 (operation) 110, 128 2-extensional 129 extensionality axiom,

see

axiom

extensionality for properties, classes 73- 74 for sets 135, 199

comprehension 58 extensional 130

family

formula 57

J~-normal 387

predicate 60

RS- 116

elimination lemma 296 embedding of admissible sets 135 theorem 190 encoding 68 of logical operators 43, 80 Engeler 13, 28, 33 enumeration axiom 27, 126, 144, 151, 152, 163 theorem 32, 42 envelope 88 equality 14, 15 definitional 403-407 extensional 28, 60, 70, 108, 110, intensional 429, 431 level 220 pointwise 113, 121, set-theoretic 132, 133

Feferman 2, 5, 6, 7, 44, 50, 55, 57, 58, 59, 66, 75, 80, 151, 152, 162, 163, 165, 166, 199, 203, 217, 226, 238, 241, 243, 253, 257, 273, 278, 402,421 field (of a relation) 126 first recursion theorem (analogue of) 104, 120 Fitch 2, 5, 43, 217, 379, 380, 386, 388, 396 Fitch's internal logic 388 Fitch's models 386 Fitch's theory 388 Fitting 70, 71, 110 fixed point axiom 50, 66 complete 47, 179 consistent 47, 179 dual 181 P-, 48

Ershov 104, 114, 121

intrinsic 186,

exact representation 108, 122

largest 50, 181

expansion (operation) 60, 78

least 48, 88, 181

explicitly CL-continuous 118

maximal 186

explicitly open 116

~0-' 357

exponential (w-) 274

~1-' 375,

exponentiation 61

~2-' 387. model 179, 186, 352,358, 379

extensional choice, s e e choice equality, s e e equality

fixed point theorem for operations 16

membership 134

for monotone operations 112

model 34

for predicates or

extensional 1 (for properties) 70, 110

properties 63

Index fixed point theory 50 Flagg 33, 39, 103, 151, 152, 156, 162,

445 28, 121 grounded element 204 Gupta 394

formula analytical 199 arithmetical 239

Halbach 385

bounded 39, 40, 138, 230, 239

Harrison 258

Harper 402

elementary 57

Hayashi 402

elementary extensional 130

Hensel 262

u-free 199

Herbrand 365,367, 372

operative 64

Herzberger 380, 390, 394

positive elementary 88

hierarchy

quasi-elementary 57 stratified 199 T-negative 53,

ramified 261, 263, 267 Veblen 270 Hilbert 15, 238

T-positive 53, 220

Hindley 18, 21, 22, 34

Fraassen, van B. 352,357

Hinman 91, 230, 233, 237, 256

Frege 1, 2,422 Frege structures 2, 43, 53 Friedman 7, 217, 227, 238,

hyperarithmetical analysis 240 hyperelementary set 87

241, 243, 258, 261, 282-83

Honsell 402

hyperjump 237, 258

379, 380, 383, 384, 394 function continuous on ordinals 269 C_-increasing 49, 181, 208 increasing on ordinals 269, 391 normal 269 uniformly A1- , ~1-' 231 fundamental sequence 275, 276 Gale 243 Gallier 282, 283 Gandy 91, 261, 262 Gilmore 73, 75, 198 Girard 7, 9, 253, 271 global consistency 222 GSdel 1, 60 GSdel numbering 42, 68, 82, 336, 338 Gordeev 73, 74 graph model 13, 28, 34 graph of a continuous operator

ideal completion 115 implication levels of, 151, 152 R- 153 incoherent element 204 independence 45, 80, 81, 164, 292,320, 407, 413 induction, s e e axiom, schema inductive definition 63-67, 85, 87, 232, 257, 351,355 model 43, 58, 85, 86, 88, 91, 97 set 88 intensional equivalence 151, 162, 174 interpretation asymmetric 7, 299, 311, 324, 325, 336 provability 369, 370, 372 intersection 59

Index

446

Levy 206

generalized 61 intrinsic element 204

liar (sentence) L 52, 204, 374, 424

intrinsic fixed point 186

local consistency 219

inversion lemma 234, 295,317,

local truth axioms 219

345,363

local truth lemma 370

involution 182

LOG- rule 353

iteration A-, 390

logic

Iterationsprinzip 217, 229

combinatory 13-16 deontic 386

J~,ger 27, 74, 144, 217, 226, 337

external 180, 399

Jockusch 262 join 58, 59, 79, 80, 223, 225,

Fitch's internal 388 Friedman-Sheard 384

226, 228, 229, 253

internal 180, 380, 394, 396, 400 modal 396

Kalmar 343

M-logic 376

Kelley 114 Kleene 3, 7, 20, 28, 80, 91, 98,

type free 6, 151, 174

104, 244, 245, 246, 248, 258, 262,343, 423 Knaster 113, 358, 424 KSnig 243 Koymans 34 Kreisel 4, 114, 238 Kripke 5, 6, 44, 139, 177, 178, 186, 203, 231,351,420, 421,423, 424 Kripke's classification 203 Kripke-Platek set theory 139 Kruskal 282

logical consequence 351,355,379, 384, 386, 388, 398 logical theory of constructions 401-403, 411 Longo 34, 35 Lorenzen 217 Lusin 243 Marek 262 Martin-LSf 217, 226, 402,403, 405,407, 419 Marzetta 74 Mc Gee 7, 380, 381,383 Meyer 17, 28

label 339

Minari 74, 112, 122, 194, 196

lambda calculus 16, 17, 18

minimalization 20

lattice 179 complete 183

Mints 312,335,338 model

involutive 182

/3-model 258, 262

non-modular 192

closed term 26

lazy evaluation 407, 408 level

Doo 35 Engeler's D-, 33

axioms 220, 290

Fitch's, 379, 386

induction, 293

fixed point, see inductive

lowering lemma 328

inductive, see inductive

of implication 151

open term 26

of t r u t h 215

w-model (w-standard) 47, 49,

Index

447

180, 194, 230, 237, 258, 358,

number-theoretic 20

388

of ordinal arithmetic 274-275,

Pw 28-32 recursion-theoretic 215, 230 recursive graph RE, 28, 32 recursively s a t u r a t e d 90

selection 127 ~- 107 operator (formula) 64

semi-inductive 394

closure 29, 36

supervaluation 357

continuous 28 CL-continuous 117

monotonicity of deducibility 153

ECL-continuous 118

of ordinal assignement 316, 362

effectively continuous 28

monotone,

see

operation, operator

Montague 383 Moschovakis 6, 8, 64, 80, 85, 88, 98, 131, 206, 207, 262,401,

RS-continuous 119 elementary 64 existential 65 monotone 29, 88, 120, 144, 207,

422 425 Musil 149 Myhill 1, 4, 6, 39, 44, 68, 71, 103, 109, 112, 147, 151, 152, 153, 156, 160, 162, 217 Myhill's theorem 71 Myhill-Shepherdson's theorem (analogue of) 112

234, 371,387 non-monotone 394 ordering connected (linear) 126, 227 directed u n b o u n d e d partial 220 onw

20

ordinal admissible 230 closure 253, 392

Nakano 402 n a t u r a l ordinal sum 318

notation 272

nesting p r o p e r t y 277

n u m b e r 145

constructive 64

NordstrSm 402

of predicative analysis F 0 271

n o r m a l form

projectible 231

Cantor 271

recursively inaccessible 231

for c o m b i n a t o r y terms 22

recursively Mahlo 255

no solution l e m m a 367 Odifreddi 82 operation basic (for predicates) 59 choice 128 combinatorial 60 extensional 2 110 K a l m a r elementary 343 lattice-theoretic 181 monotone 62, 109-110, 112, 113, 117

stabilization 392 pair 61, 77 axiom 15, 139 ordered 14 pairing combinator 21 function 42 surjective 31 paradox Curry's 53, 151 Gordeev's

74

448

Index Russell's 56

acceptable 126, 259 locally decidable 259

paradoxical combinator 17 element 204

unbounded 259 P u t n a m 262

p a r a m e t r i z a t i o n 56 quasi-well-ordering (qwo) 259

P a r k 37 Parsons 40, 41, 42 Peano 243,

see

quasi-elementary formula 57

arithmetic

period 392 periodicity 392 Perlis 420, 421 persistence axioms 220, 290, 296, 304, 314 l e m m a 298, 321

ramified hierarchy 261-263 bounded 267 n-rank 313 Rathjen 256 recursion arithmetical transfinite 240

Plotkin 13, 28, 402,419

A I - ' E l - ' 232 , 235-37 formal language of, 422

Pohlers 9, 217

on natural numbers 19

Poincar~ 7 5 , 3 4 9

on ordinals 48, 146

power set 10, 62, 161

special transfinite 260

Plato 11

prewellordering 104 primitive recursion 19, 20, 27, 39-40, 404, 409

W k - , 302 recursion theorem

first 120 second 343

principle abstraction 56

recursive functions

choice 126

partial 19, 244-45

construction 80

primitive 19, 20, 27, 39,

join 58 meta-loeb 373

40, 93, 101, 246, 303, 338, 348

reducibility 244

provably 40, 91, 101

reflection 224

representability of, 20

CL-reflection 106

recursor 19, 146, 260, 261

:~-reflection 371

reducible formula 317

process 390

m-reducible property 71

product

reducibility,

see

principle

cartesian 61

reduction l e m m a 295,318, 346

generalized 58

reduction relation 22, 23

progressive property 126, 227, 259, 354 proposition 51

infinitary 165, 166-169 reduction theorem 109 reflection 224

propositional function 51

~t~-reflection 371

provability interpretation 370

repetition rule 338

pseudo-well-ordering (pwo)

representable

126, 259

function 20

Index

449 internal abstraction 353

set 88

level transfinite induction 293

representation theorem for

local abstraction 222

extensional operations 117

CL-reflection 106

Ressayre 253 ~verse mathematics 217, 238, 241

REFL + 237

Rice 44, 68, 70, 71, 72, 110, 116,

second-order comprehension 200, 239

122 Rice's theorem 71

soundness 54

Rice-Shapiro

Tarski's 53, 385 transfinite induction 259,

family 116

278, 354

theorem 110, 122

E-transfinite induction 139

Richter 230, 232, 253, 254, 255 Rogers 70, 71, 82, 258

transfinite recursion 227-229

Russell 56, 70, 90, 151, 162, 215,

Turner's 396 type-free abstraction 56

222, 223, 351 satisfaction 68, 69

Schfitte 7, 9, 217, 241, 243, 257,

Schellinx 34

269, 270, 271, 273, 278, 280,

schema

312,316, 319

ATR 240

Schwichtenberg 302,304, 312,335, 338, 402

bar induction 241 bounded collection 139

Scott 5, 13, 17, 28, 32, 33, 34, 37,

bounded complete induction

104, 114, 217, 410

139 bounded separation 139 choice 33 A-comprehension 108 A l-comprehension 240 H~-comprehension 241 Nl-dependent choice 240 elementary choice 130

Scott's extension theorem, 32 Scott topology 28, 104, 114 section 90 Seldin 18, 21, 22, 34 selection 127 semi-inductive definition 390 model 394 sense 422 separation 98, 108,

CL-, 107

elementary comprehension 58 elementary dependent choice

sequent calculus, 286, 303-304, 314, 361

130 explicit abstraction 60 extended abstraction 222

set admissible 125, 137

generalized coinduction 207

arithmetical 240

generalized induction 87

bounded (of ordinals) 269

Herbrand 372

closed (of ordinals) 269

N-induction 15, 50

F-closed 47

=t( + )-N-induction 27

90-closed 357

Index

450 coinductive 89

supervaluation model 357

complete 47

Suslin 80, 98

consistent 47 ~-definable 262

Tait 286, 289, 314, 320, 324,

El- , Al-definable 231 A-dense 393

Tait's 2nd cut elimination 320

333, 348

F-dense 179

Takeuti 9, 238, 243

O0-dense 357

Talcott 402

02-dense 387

Tarski 37, 44, 53, 65, 70, 113, 125,

hyperelementary 89

201, 215,352,358, 385,412,

inductive 89

424

iterative 63

tautology lemma 292, 295,320, 363

representable 88

term model,

Shapiro 110, 116, 117, 122

see

model

theorem

Shepherdson 103, 109, 112

approximation 99

Shoenfield 14, 246, 376

boundedness 147

Sierpinski 243

cardinality 191

Simpson 217, 238, 241, 243, 262

characterization 376

Smullyan 109

conservation 101,335

soundness 153

decomposition 393

formalized :}r_, 370

embedding 190

positive, negative 220 A + - 221

fixed point for operations 16

~c~-' 160 splitting pair 195 stabilization theorem 391 stably inside 390 stably outside 390 n-stage 313 Stewart 243 Strahm 27 subsequence relation 244, 341 substitution closure 289 instance 289 lemma 292,316, 344 substitutivity 292, 295,320 subsystems of second order arithmetic 238-239 sum direct 61 generalized 58

fixed point for predicates 63 generalized induction 87 internal N-induction 368 Kleene basis 246 Knaster-Tarski 113 Levy absoluteness 206 Myhill-Shepherdson 112 perfect set 206 periodicity 392 reduction 107 representation 119 Rice 71 Rice-Shapiro 110 separation 107, 108 stabilization 391 Suslin-Kleene 80, 98 transfinite induction 354 tree 246 uniform ordinal comparison 94 upper bound 328

Index

451 for set 198, 239

theory admissible set 139

Vaught 201

minimal frame MF 50

Vauzeilles 253

of operations OP 15

Veblen 269, 270 Visser 194, 390

prewellordering P W 106 t r u t h with levels TL 219 VF 356 topology class 114 positive information 28 RS-topology 117 translation 140, 174, 201, 242, 247, 305, 372,413, 417 lemma 247 transpose 60 tree, recursive wellfounded 245 Troelstra 27 T-rules 361 truth reflective, self-referential 2, 5, 6, 7, 43, 44, 50, 51, 85, 103, 104, 120, 125, 151, 177, 178, 180, 196, 198, 206, 215, 216, 217, 218, 220, 223, 230, 249, 257, 258, 285, 286, 303, 311,351,379, 394, 401,403, 412, 420, 422,423, 424, 425 stable 394, 395, 396, 399 Turner 380, 394, 396 type 402, 403, 406 finite 64, 65, 75 Ulm 243 uniform ordinal comparison 94 ungrounded element 204 union 71 generalized 61 universe 61, 226 unparadoxical element 204 variable 9 individual 14 for levels 218

weakening 292, 316, 345, 363 well-founded,

tree

see

well-ordering 258-259 predicative 269, 277 Weyl 83, 213, 215, 217, 225, 229, 257, 260 Weyl's principle

229

Zorn's lemma 187

This Page Intentionally Left Blank

LIST OF SYMBOLS Part I lists the abbreviations designating formal systems, arranged in order of appearance. Part II contains abbreviations for axioms, axiom schemata and rules, while Part III contains basic abbreviations and symbols. In parts II-III, the list is arranged per chapters and, within each chapter, in order of appearance. We give the page number of the first occurrence of the each symbol we consider. I. Formal Systems PC, I, 15 OP, I, 15

classical predicate logic theory of operation

O P - , I, 16 OPA-, I, 17 CL, I, 21 PA, I, 27, 40 PRA, I, 27, 39

.... without N-induction O P - b a s e d on A-calculus pure combinatory logic Peano arithmetic primitive recursive arithmetic

PAl, I, 4O M F - , II, 43, 50 MF, II, 5O NMF, II, 54 ID1, II, 66

Peano arithmetic based on El-induction minimal framework without N-induction MF with full induction neutral minimal framework fixed point theory of elementary inductive definitions MF with class N-induction MF with property N-induction pure property theory

MFc, II, 67 MFp, II, 67 PT, II, 77 PW c ( P W - , PWp), IV, 104, 105 KPU(op), V, 139 Ec~, VI, 158 F ~- n t, VI, 158 BLc, VI, 163 MFS-, VII, 199 TL ( T L - ) , VIII, 219, 220 TLR, VIII, 224 T L R - , VIII, 224 TLR*, 250 ATR0, VIII, 241

MF c ( M F - , MFp) +approximation axioms admissible set theory above combinatory logic Myhill's system with levels of implication formal deducibility with levels of implication Behmann's logic with class-N-induction minimal framework with sets theory of truth with levels (without N-induction) theory of truth with levels and reflection TLR without N-induction reflection TLR plus axioms ONT+BLQ arithmetic transfinite recursion

Symbols

454 a L C A o, ZLAC o

1-Ii-CA o, a~2-ca0 , VIII, 241

basic subsystems of 2nd order arithmetic

MFR(p), IX, 278 STLR, X, 289

MF c plus RAM(a, p) for each a < F 0 sequent calculus for truth with levels

STLR ~176X, 294

infinitary STLR

STLRn, X, 297

sequent calculus for truth up to level < n with bounded level quantifiers

X, 297

I T n~176X, 304

union over STLRn, n E infinitary sequent calculus for truth up

RSn, XI, 314

to level n ramified system for truth of level n

STLR,

OP ~176XI, 327

OP based on w-logic

VF , XII, 352

basic non-reductive theory for self-referential

VF c (VFp), XII, 352

V F - + class (resp. property) N-induction

VF0, XII, 355

V F - in the language of pure combinatory logic

truth without N-induction

ID 1 (acc)' XII, 356

theory of accessibility inductive definitions

V F H - , VFHc~ VFH p, XII, 372 IL, XIII, 38O

extensions of V F -

FSL, XlII, 384 IFT, XIII, 388 F T ( F T - ) , XIII, 388 LIS, XIII, 396 LES, XlII, 400

internal non-reductive T-logic Friedman-Sheard system internal Fitch's logic Fitch's theory (without N-induction) internal axioms for stable truth

LTCw, XIV, 411

external axioms for stable truth logical theory of constructions (without proposition and truth predicates) LTC with propositions and truth of

LPCR, XIV, 423

lowest predicate calculus with reflection

LTC0,

XlV, 4os

arbitrary finite level

II. Axioms, rules and other symbols Chapter 1 COMB, 15

combinatory logic

PAIR, 15 NAT, 15

pairing natural numbers

NIND, 15

number-theoretic induction schema

Ext op' 16

extensionality for operations

MS.I-MS.4, 17

Meyer-Scott axioms

CT, 2O

Church's thesis

EA, 27

enumeration axiom

Symbols NIND for positive existential formulas

3(+)-NIND, 27 ACN, 33

axiom of choice restricted to N

ACN! , 33

comprehension for operations on N

EI-IND , 39

NIND for El-formulas

Chapter 2 T.1-T.5, 49

axioms for reflective truth

RES, 49 CONS, 49 COMP, 54

restriction axiom consistency axiom

AP, 56

abstraction principle

completeness axiom

EC, 58

elementary comprehension

J, 58

join principle

P-NIND, 66

property N-induction

CL-NIND, 66

class N-induction

CP, 8O

construction principle

Chapter 3 GID, 87

generalized induction schema

lr, 98

approximation operation

HAX, 100

7r-axioms (or approximation axioms)

Chapter 5 choice axiom for operations on V extensional choice axiom extensional dependent choice axiom

ACv(oP) , 128 Ext-AC, 129 Ext-DC, 129 EAC, 130 EDC, 130

elementary choice schema elementary dependent choice schema

Chapter 6 Hyp/Tnd, } Lift, D, N

axioms for Myhill's system

159

Eq, K, S IA, EA, I~A, E~A, / I V, E v , 1--1v , E-~ v , Red~ 159-160

logical rules for Myhi11's system

/

V v ; IV, EV, I~V, E~V I n D , E n D , 160 I n ' D , E n i D , 160 E.I-E.7, 164

) rules for level n implication rules for level n negated implication axioms for Behmann's logic

455

Symbols

456

Chapter 7 Set.l-Set.3, 199

set axioms

R, 199

"anti-cantorian axiom"

GID ^, 2O7

generalized coinduction principle

Chapter 8 LIND, 219

local N-induction

PRO J, 219

projectibility axiom

REF, 224

reflection principle

LIM, 226

limit axiom for universes

WP, 229

Weyl's principle

~ - C A , 239

~-comprehension schema for analysis

a~-DC, 240

~-dependent choice schema

BI, 241

bar induction

RPC, 244

reducibility principle for classes

ONT, 25O

ontological axiom

BLQ, 250

bounded level quantifier axiom

Chapter 9 RAM(p, a), IU(~), 278

278

existence axiom of bounded ramified hierarchy transfinite induction for classes of U up to /~

Chapter 10 TI(lev),

290, 293

Level induction

( ^ ), ( v ), 290

logical rules

(Vx), (3x), (Vi), (3j), 291

quantifier rules cut rule

(Cut), 291

(w), (3~), 294 (v)b, (~)b, 297

infinitary level quantifier rules

(N), ( ~ N ) , 305

rules for N

bounded level quantifier rules

Chapter 11 LOG, 314 OPER, 314 PERSij , 314 CONSi, 314 FIX i, 314 INIn, 315

logical axioms

(T a + l ) ( - ~ T a + l ) , 315

ramified successor rules for T of level n

operational axioms persistence axioms level i consistency fixed point axioms for level i initial axioms for level n

Symbols (T-LIMa), (F-LIMa), 315

ramified limit rules for T of level n

Chapter 12 T-elem, T-out, } T-univ, T-log, T-imp 352 T-rep, T-cons WF( -~ ).1, WF( ~ ).2, 356 T-Herb, 372 I-CL-NIND, 372 I-NIND, 372 T+-elem, T+-elim, ~ } T+-univ, T+-log 372 T+-imp, T+-rep T+-cons, T(T---*) ~ 372

T-schemata axioms for the largest -~-wellfounded part Herbrand's T-schema internal class-N-induction internal N-induction

strengthened T-axioms and rules strengthened T-axioms and rules

Tax-imp, T-uniVax )

Chapter 13 T-intro, T-elim, ~T-intro, ~T-elim} 388 T-negT, 388 T+-negT, 389 T-Rcomp } T-S4comp 389, 396 T-S5comp

T-rules

T~TA---~TA T(T~TA--~TA) Turner's schemata

Chapter 14 SE.1-2, 409 LE.1-LE.3, 409 NEV.1-NEV.2, 409 PT.I-PT.4ij, 411

special axioms lazy evaluation axioms number evaluation axioms axioms for propositions and truth

457

Symbols

458 III. O t h e r S y m b o l s

Introduction E[x " - t], F V ( E ) , 9 ~, W, 9 ~(X), CZ, 10 S I- A, lO Jtt~l:A, 10

F I X ( r , ~ ) , 48 0(.;1~), 48 O(21~, ~), 48 Prop(x), 51 A ^, 54

a:::~b, 54 aC:~b, 54

Chapter 1

{ x ' A } , 55 77,77,55

N,14 K , S , 14

Cl(x),55

(--,--),14 (--)1'(--)2 14 t + l , 14 ( . . . ) , (...)k, 15 '

)~xt, 15 n,15

FP, 16 R N, 19 Vn, 3n, 19 V n < m , 3n<m, 2o

CR, 23 R E D , 23 ~-- n' 23

C T M , T M , 26 en, 28 Pw, 28 RE, 28

FUN(a), 28

CL, 55 V xriy, 3xrly, 5s E ( a , f ) , 58 II(a, f ) , 58 N,-,59 N, 59 a - - b, 60

a - e b , 6o [a---,b], 61 V, 61

a| 61 I x y A ( x , y), I(A), 63 IA, 66, 67

Sat(x, y), 69 Chapter 3

C l o s A ( - ), 87

GRAPH(F), 2s

ENV(~,S),

it]]p, 30

HYP(~I~, S), 89 I N D ( .flt~, S ) , s 9

D M, 34 Doo, 34

Chapter 2

[A], 45 ID, N E G , T R , 45 ALL, A N D , OR, 45 F ( S ) , 47

SEC(.~, S),

s8

90

lal,92 7tax, 98

Chapter 4

x < zY, x < zY, 105 ~a, 105

RD(x, y), lO7

Symbols CSP(x, y), 107 SEP(x, y), 108 ER(y, z), 109

459

Chapter 7

COMP(Jtt~),

b is extensional2, 110 f is extensional I , 110 / ( f ) , 112 E ( b ) , 113 V(e), 114 Cl-Zf, 114 ECL-OPEN, 116 R S , 116 EFF, 118 ECL, 118

179

C O N S ( ~ ) , 179 F I X ( ~ ) , 179

FIXcs(Jfi~ ), 180 FIXcp(A[~ ), 180 ]~I', 180 tg~, 180 S d, 181

UP(S), DOWN(S),

D(~), 181 [.JC,

I-It,182

INT(.A,),

186

P(.At), 186

Chapter 5

MAX(Jfi~), 187 Gr(31~), 204 P a r a d o x ( . . & ) , 204

-~ w' 126

5(a, ~ ~), 12r

Ext2( f ), 128 2-Ext( f ), 129 AD(U), 132 U-AT, 132 U-SET, 132

I n t r ( J ~ ) , 204

Chapter 8 "~V, 218 i0, i 1, . . . . 218 LT, 218

- - U , ~ U, 133 m

x C uY, x C uY,

134

Vx C y, 3x C y, 138 R e ( f ) , 147

~,--/,218

Tit, Fit, 218 tr]is, t-~is, 218 Cli(t),218 A +, 218

Chapter 6

R, R(i), 222

FFRa,

153

x~:~i y, 222

Adr(R),

153

Univ(y),

TR(y, A, -~ w, z),

rt

A-

B, 162

Funcl(f), :::~, 166

RT(r), 167 LR, 16s --~ 169 :::>c~' 169 C~ ~

226

T I ( -~ w, b), 22r

RD,154 D , 157

166

227

La, 230 ~1 (L~), A l(Lc~), 231 /~-~-, 232 Ta, 232 t , 232 IN, 232

v(~),

235

Ct, 237

181

Symbols

460

NF(A),

"~2' 239 A0, ~10, II 1, 239 ~,

293

C~#j3 , 295 (see ch.ll)

II~, II1, ~ , 239

S T L R ~ 1 7 I6- ~ F, 294

"~'n* ' 297

T Pe , 245

A[m,n], rim, n],

W(X),245

c~ F, 299

~Jc~(P), 245

LevPar(F), 299

Chapter 9

f: CL~CL,

[ r I, 299

ffi~( w k ) , 302

259

aj~a, 262

POS n and NEGn,

W~k, 262

Kn(A),303 en, 306

GO(E),

264

U"(z) 267 Cg

Chapter ll

f" ~ ~ ~, 269 E X , 269

~'n, r' 312

fix(f),

T~(t), Fa(

269

Lev(B), 313 Stn(A),313

270

Rn(A), 313 Ac~ , 313

CO, 27O r o, F ~ , 270

~'Y, 270 f ( t l , s 1) . . . . .

CN,

RSn~

f ( t n,sn), 272

272

, 272 t[,~], 276

Good(%P(z)), 278 IU(fl),

, 313

toC~s, t~us , 313

E f ix( f ), f' ,269

r

303

278

Chapter 10 ~V+~ 288

A+,E,

II, A o, 287 Lc( A ), 288

rk(A), 288 Ti_Clause(t), Fi-Clause(t ), 289

p F~ 316

~#j3, 318 A[fl,7], 324 O P ~ F- p

F~ 327

TI(~), 335 TI(

< 7), 335

T I o p ( < ol), 335 [ E] , 336

Dimk(d, [A1, ~r), 336 Truen(FA]), 336 [e](x), 338

RF(f) , END(f), 339 LAT(I), A F ( / ) , 339 DEP(I), 339 CODE, 340

Symbols OT*,

341

Length(s),

C h a p t e r 14

341

DEn(f),342

OB, 403 BOOL , 403

LC(DER(f)),342 f t - pc~ F, 344 CF( f ), 347

(~----~fl), ( O q . . . O~n)---+Ol , 403 _J_, 403

~TC, 9 C h a p t e r 12

IMPLY, 352 WF( -~ ), 354 XII-A, 357 SENT(JtI~), 357 F I X o ( . A g ), 357

Ate(a),

Eats(a),

Sentcc(a),

359

ON, 360 Ord(x), 360 AD, 360 TI(ON, B),

E-

E ' : or, 4o5

[E]o. , 406 m o . , 406

N ( x ) , 409

360

Nat(x), 410 EqNat(x, y), 41o

C h a p t e r 13 383

T h oo(.J~ ),

403

(xy)f(x, y), 404 LEV, 404 NEV, 404 Pi(x), T, 404 Pa, Deeidenat, 404 Spread, Decide, 404 inl(x), inr(x), Ind, 404 -- o" 406

}__p a F=~/k, 361

J(X),

461

383

ko(A), 383 Th(,At~), 383 Diag(~), 386 .At-NOR, 3s6 X I I - 2 A, 387 (I)2 ( X ) , 387

N~(.AI~), 387

limin f , 39o In(A, X ) , In(X),

390

Out(A, X), Out(X),

Stab(A, X), 390 Unstab(A, X), 390 Con f (X), 393 Cycle(X), 393 Init(X), 393

390