Michael 1. Beeson

Foundations of Constructive Mathematics

Metamathematical Studies

Springer-Verlag Berlin Heidelberg NewYork Tokyo Michael J. Beeson Department of Mathematics and Computer Science San Jose State University San Jose, CA 95192, USA

AMS Subject Classification (1980): 03F50, 03F55, 03F60, 03F65

ISBN-13: 978-3-642-68954-3 e-ISBN-13: 978-3-642-68952-9 DOl: 10.1007/978-3-642-68952-9

Library of Congress Cataloging in Publication Data Beeson, Michael J., 1945-. Foundations of constructive mathematics. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 6) Bibliography: p. Includes indexes. I. Constructive mathematics. 1. Title. II. Series. QA9.56.B44 1985 511.3 84-10583 ISBN 0-387-12173-0 (U.S.)

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illus· trations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1985 Softcover reprint of the hardcover 1st edition 1985 Typesetting, printing and binding: Universitiitsdruckerei H. Stiirtz AG, 8700 Wiirzburg 2141/3140-543210 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . Band 6 A Series of Modem Surveys in Mathematics

Editorial Board E. Bombieri, Princeton S. Feferman, Stanford N.H. Kuiper, Bures-sur-Yvette P. Lax, NewYork R Remmert (Managing Editor), Miinster W. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris This book is dedicated to the memories of Errett Bishop and Arend Heyting Table of Contents

Introduction XIII User's Manual XVIII Common Notations XX Acknowledgements XXII

Part One. Practice and Philosophy of Constructive Mathematics 1

Chapter I. Examples of Constructive Mathematics 3 1. The Real Numbers 3 2. Constructive Reasoning . . . . . 5 3. Order in the Reals ...... 6 4. Subfields of R with Decidable Order 8 5. Functions from Reals to Reals 9 6. Theorem of the Maximum . 10 7. Intermediate Value Theorem 11 8. Sets and Metric Spaces 12 9. Compactness ...... 13 10. Ordinary Differential Equations 14 11. Potential Theory . 15 12. The Wave Equation . 17 13. Measure Theory 17 14. Calculus of Variations 19 15. Plateau's Problem 20 16. Rings, Groups, and Fields 22 17. Linear Algebra . . . . 24 18. Approximation Theory 25 19. Algebraic Topology . . 26 20. Standard Representations of Metric Spaces 27 21. Some Assorted Problems ...... 30

Chapter II. Informal Foundations of Constructive Mathematics 33 1. Numbers 33 2. Operations or Rules 34 3. Sets and Presets 34 VIII Table of Contents

4. Constructive Proofs 35 5. Witnesses and Evidence 36 6. Logic ..... 38 7. Functions ...... 40 8. Axioms of Choice 42 9. Ways of Constructing Sets 43 10. Definite Presets .... 45

Chapter III. Some Different Philosophies of Constructive Mathematics 47 1. The Russian Constructivists 47 2. Recursive Analysis 48 3. Bishop's Constructivism 49 4. Objective Intuitionism . 49 5. Sets in Intuitionism . . 52 6. Brouwerian Intuitionism 52 7. Martin-Lars Philosophy 53 8. Church's Thesis 55

Chapter IV. Recursive Mathematics: Living with Church's Thesis 58 1. Constructive Recursion Theory ...... 59 2. Diagonalization and" Weak Counterexamples" 60 3. Continuity of Effective Operations .... . 61 4. Specker Sequences ...... 66 5. Failure of Konig's Lemma: Kleene's Singular Tree 67 6. Singular Coverings ...... 68 7. Non-Uniformly Continuous Functions 70 8. The Infimum of a Positive Function . 71 9. Theorem of the Maximum Revisited . 72 10. The Topology of the Disk in Recursive Mathematics 73 11. Pointwise Convergence Versus Uniform Convergence 76 12. Connectivity of Intervals ...... 77 13. Another Surprise in Recursive Topology ...... 77 14. A Counterexample in Descriptive Set Theory . . .. 79 15. Differential Equations with no Computable Solutions 80

Chapter V. The Role of Formal Systems in Foundational Studies 82 1. The Axiomatic Method 82 2. Informal Versus Formal Axiomatics ...... 83 3. Adequacy and Fidelity: Criteria for Formalization 84 4. Constructive and Classical Mathematics Compared 87 5. Arithmetic of Finite Types ...... 88 6. Formalizing Constructive Mathematics in HAW 89 Table of Contents IX

Part Two. Formal Systems of the Seventies ...... 93

Chapter VI. Theories of Rules 97 1. The Logic of Partial Terms 97 2. Combinatory Algebras 99 3. Axiomatizing Recursive Mathematics 107 4. Term Reduction and the Church-Rosser Theorem 111 5. Combinatory Logic and A-Calculus 115 6. Term Models ...... 119 7. Continuous Models ...... 127 8. Finite Type Structures and Continuity in Combinatory Algebras 134 9. Set-Theoretic and Topological Models . . . 143 10. Discussion: Adequacy and Fidelity of EON? ...... 145

Chapter VII. Realizability ...... 148 1. Definition and Soundness of Realizability 148 2. Realizability and Models 152 3. Some Simple Applications 152 4. Existence Properties 156 5. q-Realizability . . . . . 157 6. Rules of Choice .... 158 7. Discussion: Numerical Meaning 159

Chapter VIII. Constructive Set Theories 162 1. Intuitionistic Zermelo-Fraenkel Set Theory, IZF 162 2. Non-Extensional Set Theory ...... 166 3. The Double-Negation Interpretation for IZF . 173 4. Realizability for Set Theory Without Extensionality 176 5. Realizability and Models ...... 184 6. Realizability for IZF ...... 185 7. Connection with Realizability for Arithmetic 187 8. Consistency of Church's Thesis with IZF . 188 9. The Numerical Existence Property for IZF 188 10. Discussion . . . 190 11. The Theory B ...... 193 12. More Discussion ...... 196 13. Intermediate Constructive Set Theories 199

Chapter IX. The Existence Property in Constructive Set Theory 202 1. Introduction ...... 202 2. The Set Existence Property for HAS . 203 3. The Existence Property in Set Theories 206 x Table of Contents

Chapter X. Theories of Rules, Sets, and Classes . 216 1. The Theory FML ...... 216 2. The Theories EMo and EMo + J 218 3. Models of Feferman's Theories 221 4. Realizability . . . . 225 5. The Axiom of Choice . 227 6. q-Realizability . . . . 229 7. Term Existence Property 232 8. Evaluation of Numerical Terms 232 9. Numerical Existence Property 233 10. Decidable Equality . . . . . 233 11. Extensionality in Feferman's Theories 235 12. Some Remarks on Formalizing Mathematics in FML+DC 236 13. Functions, Operations, and Axioms of Choice 238 14. The Theory SOF (Sets, Operations, Functions) 239 15. Some Theorems of SOF 239 16. Realizability for SOF 240 17. Discussion ...... 241

Chapter XI. Constructive Type Theories ...... 246 1. Forms of Judgment ...... 246 2. Philosophical Remarks on Sets, Categories, and Canonical Elements 248 3. Hypothetical Judgments ...... 249 4. Families of Sets ...... 250 5. Disjoint Union and Existential Quantification 250 6. Abstraction ...... 251 7. Cartesian Product and Conjunction . . . 251 8. The Constant E and Projection Functions 252 9. Product and Universal Quantification 253 10. Implication 254 11. N, Nk , and R . 254 12. Disjoint Union 254 13. The I-Rules 255 14. How Martin-Lars Rules Actually Define a Formal System 255 15. List of Rule Schemata of Martin-Lars System MLo 257 16. Other Formulations of MLo ...... 261 17. Interpretation of HAOl+AC + EXT in MLo . 263 18. Formalizing Mathematics in MLo . . . . . 265 19. Is the Theory HAOl + EXT + AC Constructive? 267 20. The Realizability Model of MLo ...... 268 21. Martin-Lars Universes ...... 275 22. The Arithmetic Theorems of Martin-Lars Systems, and a New Realizability for Arithmetic 277 23. Discussion ...... 279 Table of Contents XI

Part Three. Metamathematical Studies 285

Chapter XII. Constructive Models of Set Theory 287 1. Aczel's" Iterative Sets" ...... 287 2. Interpreting Subtheories of Intuitionistic ZF in Feferman's Theories 296

Chapter XIII. Proof-Theoretic Strength . . . . 300 1. Definitions About Proof-Theoretic Strength 300 2. L'~ -AC and ID! ...... 303 3. The Strength of Feferman's Theory EMo +J and Related Theories 309 4. The Strength of Constructive Set Theory T 2 310 5. The Strength of Martin-LoPs Theories . 316 6. Theories with the Strength of Arithmetic 318 7. Conservative Extension Theorems . . . 320

Chapter XIV. Some Formalized Metamathematics and Church's Rule 324 1. The Theories Tb and Ta ...... 324 2. Formalization of Normal-Term Arguments 327 3. Formalized Models ...... 331 4. Church's Rule 332 5. Truth Definitions and Reflection Principles 337 6. Formalized Realizability and Existence Properties 341 7. Discussion ...... 344

Chapter XV. Forcing 346 1. Ordinary Forcing ...... 347 2. Conservative Extension Results 355 3. Uniform Forcing . . 359

Chapter XVI. Continuity 368 1. Continuity Principles . . . . 369 2. Continuity and Church's Thesis 380 3. Consistency of Brouwer's Principle and Uniform Continuity 385 4. Derived Rules Related to Continuity ...... 390

Part Four. Metaphilosophical Studies 399

Chapter XVII. Theories of Rules and Proofs 401 1. Review of the Relevant Literature 401 2. The Main Issues ...... 402 3. A Theory of Rules and Proofs 403 XII Table of Contents

4. Undecidability of the Proof-Predicate in C 404 5. Consistency of C 405 6. Discussion ...... 407 7. Frege Structures ..... 410 8. Existence of Frege Structures 411 9. Set Theory and Frege Structures 412 10. A Theory of Rules, Proofs, and Sets 415

Historical Appendix ...... 417 1. From Gauss to Zermelo: The Origins of Non-Constructive Mathematics 418 2. From Kant to Hilbert: Logic and Philosophy 424 3. Brouwer and the Dutch Intuitionists ...... 430 4. Early Formal Systems for Intuitionism ...... 432 5. Kleene: The Marriage of Recursion Theory and Intuitionism 433 6. The Russian Constructivists and Recursive Analysis 434 7. Model Theory of Intuitionistic Systems 435 8. Logical Studies of Intuitionistic Systems 435 9. Bishop and his Followers 437 10. The Latest Decade 438

References 439 Index of Axioms, Abbreviations, and Theories 451 Index of Names . 455 Index of Symbols 459 Index 461 Introduction

This book is about some recent work in a subject usually considered part of "logic" and the" foundations of mathematics", but also having close connec• tions with philosophy and computer science. Namely, the creation and study of "formal systems for constructive mathematics". The general organization of the book is described in the" User's Manual" which follows this introduction, and the contents of the book are described in more detail in the introductions to Part One, Part Two, Part Three, and Part Four. This introduction has a different purpose; it is intended to provide the reader with a general view of the subject. This requires, to begin with, an elucidation of both the concepts mentioned in the phrase, "formal systems for constructive mathematics". "Con• structive mathematics" refers to mathematics in which, when you prove that a thing exists (having certain desired properties) you show how to find it. l Proof by contradiction is the most common way of proving something exists without showing how to find it - one assumes that nothing exists with the desired properties, and derives a contradiction. It was only in the last two decades of the nineteenth century that mathematicians began to exploit this method of proof in ways that nobody had previously done; that was partly made possible by the creation and development of set theory by Georg Cantor and Richard Dedekind. The abstract notions of set theory, especially the system• atic consideration of infinite sets, provided a nutrient broth in which mathemat• ics grew by leaps and bounds. Certain mathematicians were suspicious of these abstractions, especially the use of infinite sets as "completed". Their suspicions seemed at least partly justified when several "paradoxes" were discovered, in• volving things like "the set of all sets which are not members of themselves", to mention just one. 2 Naturally enough, a reaction set in: although the "liberal" set-theorists had opponents all along, the most vigorous spokesman by far for a compara• tively "conservative" philosophy and practice of mathematics was the Dutch mathematician L.E.1. Brouwer. 3 His ideas are known as "intuitionism", proba• bly because he emphasized the role of intuition in recognizing that a mathe• matical proof is convincing, as opposed to the" formalist" idea that a mathe• matical proof is just a sequence of signs which follow certain rules.

1 The first three chapters ofthe book are devoted to explaining in more detail, with the necessary distinctions and subtleties, what constructive mathematics is. 2 There is an appendix to this book in which I have made an effort to provide some historical background. Originally that appendix was intended as an introduction, but it grew too detailed. 3 As is well known, Brouwer also did pioneering work in topology. XIV Introduction

There ensued a long debate, in which Brouwer's most forceful opponent was the German mathematician Hilbert. The issue was (or at least seemed to be): power versus security.4 Hilbert spoke glowingly of the set-theoretical "paradise that Cantor has created for us", and said that depriving a mathemati• cian of the principle of proof by contradiction was like depriving a boxer of his fists. Brouwer rejected both the paradise and the fists that were needed to defend it, thereby ensuring that whatever mathematics remained was free of paradox and completely reliable. As is usual in human affairs, the drive to power was stronger than the drive to security, and by the 1960's, when the author went to graduate school, Brouwer was not even mentioned to most students in mathematics. The controversy was over. Set theory was taught as early as second grade and existence proofs by contradiction were taken for granted. So much for a brief perspective on "constructive mathematics". I also pro• mised you an elucidation of the phrase" formal systems". This concept, which is the centerpiece of modern mathematical logic, goes back at least to Leibniz, who in the sixteenth century dreamed of a universal language in which all mathematical and philosophical ideas could be precisely expressed. There would be rules for manipulating the symbols, and if there were disagreement, you would not have to engage in long and fruitless argumentation - you would say, "Sire, let us calculate". In 1854, George Boole took the first steps in this direction by inventing a system of symbols and rules for what is now called "propositional logic" - logic with "and", "or", and "not". Later in the nineteenth century, Frege and Peano constructed more elaborate and exten• sive systems of symbols and rules for expressing logical and mathematical argu• ments. The continued refinement and study of these systems has been the subject matter of logical investigations throughout the twentieth century.5 In particular, systems were developed which provided symbols and rules for manipulating them which correspond to the kinds of mathematical reasoning allowed by the intuitionists. These systems were set up and studied by Heyting and Kleene, among others; from a logical standpoint, they turned out to have much in common with the systems for classical mathematics, although in some ways their metamathematics is more subtle. 6 In 1966, then, the subject of constructive mathematics was not a hotbed of activity. In the 1970's, however, there was a flurry of new research, which

4 Although "Hilbert's program" was concerned with security, it existed because Hilbert wanted to recover the security he believed was lost by refusing to leave Cantor's paradise and give up the law of the excluded middle. One of Brouwer's first publications was about the unreliability of the logical laws, which indicates that this was one of his main concerns. 5 Incidentally, the investigations of K. G6del show that Leibniz's original dream can never come true, even if we restrict the system to questions about the arithmetic operations on the natural numbers. Whatever system of rules Leibniz's successors might make up, G6del could construct a question not answerable by calculation using those particular rules. 6 One of the main forces behind the development of formal systems has always been the drive to rigor, the desire to see every step filled in and justified. Whether that helped or hindered in the quest for security alluded to above seems to have been a matter of opinion: the formalists thought it helped and Brouwer (and some of his predecessors such as Borel and Poincare) thought it hindered. Nevertheless, formal rigor is possible on both sides; that is, one can write down formal systems codifying the kinds of arguments actually used in proofs. Introduction xv it is the purpose of this book to systematically describe. In my opinion, there are two principal causes of this renaissance: Bishop and the computer. During the 1960's the computer was used for the numerical solution of certain mathe• matical problems, perhaps most dramatically for the computation of orbits for spacecraft. This stimulated considerable mathematical development in nu• merical analysis and numerical algebra, as people developed suitable algorithms and studied their behavior.7 Propagating outwards from these mathematicians and engineers was a renewed interest in the kind of solution to a mathematical problem which enables one to compute the solution. The result was inevitably a re-examination of the questions that had been debated by Brouwer and Hil• bert. This re-examination was undertaken by Errett Bishop in California. His book, Foundations of Constructive Analysis, was published in 1967. 8 It was vigorously discussed by those mathematicians and logicians who had come in contact with Bishop or his work, and later by a widening circle. The thrust of Bishop's work was that both Hilbert and Brouwer had been wrong about an important point on which they had agreed. Namely, both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis). Bishop showed that this was simply false, and in addition that it is not necessary to introduce unusual assumptions that appear contradictory to the uninitiated. The perceived conflict between power and security was illusory! One only has to proceed with a certain grace, instead of with Hilbert's "boxer's fists". 9 Bishop found the way to proceed by thinking of mathematics as a "high-level programming language", in which proofs should be written. Algorithms should then naturally accompany existence proofs. Bishop was not alone in this view. One may mention Martin-LOfin Sweden,

7 The atomic bomb was designed using roomfuls of human beings to perform the computations. The algorithms for solution to differential equations, such as the overrelaxation method for Laplace's equation, also were originally executed in the early fifties by roomfuls of people, each programmed for a simple repetitive calculation, taking input from and providing output to other human calculators. So a case can be made that the impetus to algorithm development came at least in part from the problems that the overall advancement of technology required to be solved, rather than from the existence of the computer. S It may be objected that (a) such a re-examination was not inevitable, or even (b) the re• examination that did take place was not in fact the result of developments connected with the computer, but arose naturally from Bishop's concern with the philosophy of mathematics. Evidence for (a) consists in the fact that numerical analysts are largely ignorant of systematic constructive mathematics, but instead treat problems case-by-case. In spite of the obvious difficulties in thoroughly refuting these objections, it does seem to the author that the increasing concern with numerical calculations must inevitably (eventually) have spread through chains of conversations and papers and exerted an indirect influence on those who concerned themselves with foundations of mathemat• ics. 9 There may well have been people who were aware of this point before Bishop's book. One expert wrote to the author, .. all competent people were aware of the scope of constructive mathemat• ics in the fifties." This group may have included the few intuitionists in Holland and a few proof• theorists elsewhere. If any of these people knew that a substantial body of constructive mathematics could be done without appearing to contradict classical mathematics, no written evidence of it survives. See Fraenkel and Bar-Hillel [1958], pp. 253-264, for a detailed statement of what was known in 1958 about the extent of intuitionistic mathematics. XVI Introduction and de Bruijn in The Netherlands; the former was creating a philosophy of mathematics based on sets as data structures and proofs as programs; the latter was actually implementing a computerized proof-checker (AUTOMATH) based on similar ideas. But Bishop's book directly stimulated most of the work de• scribed in this book, except for Martin-Lors; and it has some relationship even to that. The work in question consists in the creation and study of formal systems capable of reflecting the style of mathematics in Bishop's book. Such systems were created by Feferman, Friedman, Myhill with this purpose in mind (and by Martin-Lof for his own purposes). These systems and the results pro• duced in a decade of metamathematical studies of them are the subject of this book. It was already mentioned that the subject matter lies somewhere between logic, mathematics, philosophy, and computer science. Strands from each of these are visible in the work of the people just mentioned: Bishop emphasized the mathematics; Martin-LOf the philosophy; de Bruijn the computer; and Fe• ferman, Friedman, and Myhill the logical systems. Each of these facets of the subject is visible in the book, and the subject should continue to link these four disciplines. One way to indicate the connections is to make a list of the sorts of people to whom this book may be of interest: Mathematicians interested in constructive mathematics and in the philosophy of mathematics generally. Mathematicians and computer scientists whose work involves the study of algo• rithms, either of particular algorithms or algorithms in general. Computer scientists interested in the extraction of algorithms from proofs. Graduate students of logic or computer science wishing to study the part of logic bordering on computer science. Proof-theorists and other experts on the subjects treated here. Logicians not expert in the subject, but interested in the metamathematics offormal systems or in constructive mathematics and its philosophy. Computer scientists wishing to know more about logic, especially constructive logic, which is perhaps the natural logic of computation. Computer scientists wishing to implement practical systems for man-machine communication about mathematics. Philosophers interested in the philosophy of mathematics and with the relevant training in logic and mathematics. Having spent so many words on the origins of the subject, I may be permitted a few speculations about its future. I heard the mathematician Prof. C. Truesdell give a lecture in Milan in which he suggested that the computer may have an impact on mathematics comparable to that which the microscope had on biology and the telescope on astronomy. This may well be an overstatement, but I think it is evident that the computer will be increasingly influential in mathematics. The formal systems that have so far been developed were devel• oped in order to study the possibilities in principle of reducing reasoning to calculation. There was never a serious intent to actually use these systems. 10

10 Except perhaps in the beginning. Boole wanted to use his systems, Peano did use his, and of course there was Principia Mathematica. But even these examples had the character of demonstra• tions that the system could in principle be used to codify actual reasoning in mathematics. Introduction XVII

The existence of the computer introduces a fundamental change, by making possible symbol manipulation on the necessary scale. It is now possible to create working formal systems in which one can actually write mathematics. This is a task for the logicians and computer scientists of the eighties and nineties. It would surely happen even if pure intellectual curiosity were the only motiva• tion. As it happens, however, this is not the case. The impact of the computer on other aspects of our society requires the development of reasoning machines insofar as this turns out to be possible. Making the computer reason will eventu• ally require elaborate languages or systems which will be the descendants both of today's programming languages and of today's logical languages. Already the program MYCIN at Stanford University Medical Center uses logic to make medical diagnoses on which people's lives may depend. As the computer takes over more and more complicated tasks, it is more and more necessary and important that it be given the capability to explain its reasons. Using the algebra• ic equation solver and symbolic integration program MACSYMA, for instance, one wishes it could provide a proof that the given equations have no solution, rather than a mere announcement! One would also like the computer that warns us of an impending missile attack to be articulate and logical about its reasons; after all, last year it made several mistakes. When suitable languages are constructed (for a particular purpose, be it calculus, mathematics in general, or handling and reasoning about census data or library catalogues), then the relationship between proofs and algorithms should be very close. The natural logic to be built into such systems is constructive. That's why I see the formal systems studied in this book as "pilot studies" towards the systems of the future.

June 1984 Michael Beeson User's Manual

The initial portions of this book are meant to be accessible to anyone with advanced undergraduate training in mathematics, particularly including theo• retical analysis and advanced algebra. To be in a position to read the entire book seriously, recommended background also includes the following: (i) Some constructive mathematics: the first four chapters of Bishop's book. (ii) Some logic: at least a course in mathematical logic through Godel's theo- rem. (iii) Some familiarity with intuitionistic logic and arithmetic. A good prepara• tion would be Troelstra's article [1977H] (except for the material on choice sequences). The book is organized into four Parts: 1. Practice and philosophy of constructive mathematics 2. Formal systems of the seventies 3. Metamathematical studies 4. Metaphilosophical studies Each Part has its own introduction; there the contents of the book are explained in more detail than in the Introduction to the book. The exercises are a repository of interesting and useful facts which could not be included in the main text. Some of them are intended to elaborate a point or familiarize the student with an idea that is explained in the text; but many are vehicles for the presentation of extra results. These are of several kinds: some of them (perhaps the majority) are fairly routine; a few are routine provided you know a little more logic than is officially required (for example Kripke models). Still others require a lot of time and/or paper, even though they are fairly straightforward (for example, verifying that certain theorems can be constructively proved). Finally, there are some exercises which are actual• ly research problems. I have marked these with an asterisk, leaving it to the reader to discover which kind the others are. On page XX there is a section called "Common Notations" which ex• plains something about our use of symbols; consult it before beginning to read seriously. There is also an index of symbols which will hopefully enable the reader to find the definition of any mysterious symbols encountered. There is a separate index of names of formal systems and axioms. I have made a deliberate effort to make it possible to start reading the book at any point, limited only by one's knowledge of the subject matter and not by notation introduced in a previous chapter. Especially the chapters in Part Two (where the main formal systems are explained) are meant to be independent of each User's Manual XIX other insofar as possible, in order to make it possible to use the book as a reference work on these systems. But the serious student will find that there are many small ways in which ideas of previous chapters are built upon, and should therefore read the book in order. The one exception to this is that the Historical Appendix may be read at any time, depending on the reader's interest in historical matters. This book attempts both to present and to evaluate critically the formal systems presented in Part Two and the studies of them in Parts Three and Four. Since there is by no means a consensus about the significance of these systems, the author has allowed a whole cast of characters to present various (often conflicting) views. Some discussions between these characters take place at the end of all but the most innocuously technical chapters from Chapter VI on, and occasionally even in the middle of a chapter. The characters are intro• duced in the introduction to Part II. The reader with nothing better to do (after working all the exercises) can pass half an hour by comparing the views of the characters with the people listed in the Acknowledgements, to try to discover whose ideas have been combined and distorted almost beyond recogni• tion before being placed in the mouth of one of the characters. (There is an occasional hint when someone is directly quoted.) Common Notations

Here we list some common notations that are used throughout the book. This is not intended to be a complete list; if mystified by a symbol while reading the book, use the index of symbols, not this section.

1. Some Commonly Occurring Sets

N is the set of natural numbers 0,1,2, ... N (x) is a unary predicate in several formal theories, which in the intended model is interpreted to mean that x belongs to N. R is the set of real numbers. Q is the set of rational numbers. R + is the set of positive real numbers; similarly for Q +. C(X, Y) is the space of functions from X to Y that are continuous, and uniformly continuous on compact subsets of X. In this context usually X and Yare complete separable metric spaces. NN is the space of functions from N to N; 2N is the space of functions from N to {O, I}.

2. Sequence Numbers and Pairing Functions

We use the standard logical notation, including

Another common notation is l(n) =

3. Recursion Theory

T(e, x, n) means e is a (number coding a) Turing machine, and n is a (code of a) computation by e at input x. U is the result-extracting function, so that if T(e, x, n), then U(n) is the result of the computation n. The e-th partial recursived function is denoted by {e}(x), or sometimes by cPe(x). As usual, J1 is the least-number operator; we have

{e}(x) = cP.(x) = U(J1nT(e, x, n)).

We use L:~ and II~ for the arithmetical hierarchy; thus a predicate is II? if it has the form V xR(x, y) where R is primitive recursive; L:? if it has the form j xR(x, y) where R is primitive recursive; II~+ 1 if it has the form V xR(x, y) where R is L:~; and L:~+ 1 if it has the form j xR(x, y) where R is II~. Simi• larly, we define II~ and L:~ formulae (of arithmetic); x and yare allow to be lists of several variables. {e}f is the e-th function recursive in f, defined e.g. by means of Turing machines with oracles. Acknowledgements

I could never have written this book all by myself; in order to keep at the task in the face of other pressing obligations I needed the encouragement of my colleagues. Instrumental in providing that encouragement when I needed it were D. van Dalen, S. Feferman, and M.J. Greenberg. Countless people helped me by discussing the subject, and that in rather widespread places. I have discussed the subject of this book north of the Arctic circle in Norway, near the equator in India, in eight European countries and across the United States. That would never have been possible without the assistance of those who invited me here and there. Several of those same people (and in one case, their students) read one or more chapters of the book and found points that needed amplification, clarification, revision, or outright correction. The histori• cal appendix in particular underwent several drafts, each time benefitting from the collective knowledge of those who were asked to criticize it. Since so very many people have been involved in this project, it was not feasible to give the details of my indebtedness to each one. Their real thanks must be the satisfactory appearance of the parts of the book which most concern them; but I want the readers of this book to know that I appreciate the help of the following people: P. Aczel, F. Arzarello, H. Barendregt, J. Bergstra, R. Brummelhuis, D. van Dalen, H. Edwards, S. Feferman, R. Grayson, M.J. Greenberg, N. Good• man, S. Hayashi, J.J. de longh, H. Jervell, P. Martin-Lor, G. Renardel, P. Rodenburg, A. Scedrov, D. Scott, G. Stolzenberg, G. Sundholm, H. de Swart, A.S. Troelstra, W. Veldman, and A. Visser. This book would also not have been possible without the financial support I received during the time that I wrote it. For this I am principally indebted to the scientists and taxpayers of The Netherlands, especially A.S. Troelstra (who invited me to Amsterdam), D. van Dalen (who invited me to Utrecht), J.J. de longh (who invited me to Nijmegen), and H. de Swart, who was instru• mental in persuading the Dutch National Science Foundation (Z.W.O.) to sup• port me for eight months in addition, for which I am most grateful both to him and to the Z.W.O. A major portion of the book reached first-draft form during that period. This was a recalcitrant book, and it refused to get finished for some time after I had left The Netherlands. My new department chairman and dean, J. Mitchem and L. Lange, at San Jose State University in California, were quite supportive and released me from teaching one course to help me finish the book. To all of these people go my heartfelt thanks. It has been a pleasure to work with the professionals on Springer-Verlag's editorial and production staffs. Acknowledgements XXIII

Finally, it is a pleasure to acknowledge my appreciation to the editors of the Ergebnisse series for including my book. lowe a special debt to Solomon Feferman, who has encouraged me and contributed to my work not only in his role as Ergebnisse editor, but as mentor and colleague as well.