Introduction to Metamathematics

560 Pages·1971·14.497 MB·English

by  Stephen Cole Kleene

INTRODUCTION TO METAMATHEMATICS
BIBLIOTHECA MATHEMATICA
A Series of Monographs on Pure and
Applied Mathematics
Volume I
Edited with the cooperation of
THE „MATHEMATISCH CENTRUM”
and
THE „WISKUNDIG GENOOTSCHAP”
at Amsterdam
Editors
N. G. de Bruijn
J. de Geoot
A. C. Zaanen
INTRODUCTION TO
METAMATHEMATICS
BY
STEPHEN COLE KLEENE
PROFESSOR OF MATHEMATICS AT THE UNIVERSITY
OF WISCONSIN (MADISON, WIS., U.S.A.)
WOLTERS-NOORDHOFF PUBLISHING - GRONINGEN
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • OXFORD
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
©WOLTERS-NOORDHOFF PUBLISHING
AND NORTH-HOLLAND PUBLISHING COMPANY, 1971
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, photocopy­
ing, recording or otherwise, without the prior permission of the Copyright owner.
Library of Congress Catalog Card Number 70-97931
North-Holland ISBN 0 7204 2103 9
American Elsevier ISBN 0 444 10088 1
First published 1952
First reprint 1957
Second reprint 1959
Third reprint 1962
Fourth reprint 1964
Fifth reprint 1967
Sixth reprint 1971
Seventh reprint 1974
Publishers:
WOLTERS-NOORDHOFF PUBLISHING - GRONINGEN
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • OXFORD
Sole distributors for the U.S.A. and Canada:
AMERICAN ELSEVIER PUBLISHING COMPANY, Inc.
52 Vanderbilt Avenue
New York, N.Y. 10017
Printed in The Netherlands
PREFACE
Two successive eras of investigations of the foundations of mathematics
in the nineteenth century, culminating in the theory of sets and the
arithmetization of analysis, led around 1900 to a new crisis, and a new
era dominated by the programs of Russell and Whitehead, Hilbert and of
Brouwer.
The appearance in 1931 of Godel’s two incompleteness theorems,
in 1933 of Tarski’s work on the concept of truth in formalized languages,
in 1934 of the Herbrand-Godel notion of 'general recursive function’,
and in 1936 of Church’s thesis concerning it, inaugurate a still newer
era in which mathematical tools are being applied both to evaluating the
earlier programs and in unforeseen directions.
The aim of this book is to provide a connected introduction to the
subjects df mathematical logic and recursive functions in particular, and
to the newer foundational investigations in general.
Some selection was necessary. The main choice has been to concentrate
after Part I on the metamathematical investigation of elementary number
theory with the requisite mathematical logic, leaving aside the higher
predicate calculi, analysis, type theory and set theory. This choice was
made because in number theory one finds the first and simplest exempli­
fication of the newer methods and concepts, although the extension to
other branches of mathematics is well under way and promises to be
increasingly important in the immediate future.
The book is written to be usable as a text book by first year graduate
students in mathematics (and above) and others at that level of mathe­
matical facility, irrespective of their knowledge of any particular mathe­
matical subject matter.
In using the book as a text book, it is intended that Part I (Chapters
I — III), which provides the necessary background, should be covered
rapidly (in two or three weeks by a class meeting three times a week). The
intensive study should begin with Part II (Chapter IV), where it is es­
sential that the student concentrate upon acquiring a firm grasp of
metamathematical method.
The starred sections can be omitted on a first reading or examined in
VI PREFACE
a cursory manner. Sometimes it will then be necessary later to go back and
study an earlier starred section (e.g. § 37 will have to be studied for § 72).
Godel’s two famous incompleteness theorems are reached in Chapter
VIII, leaving a lemma to be proved in Chapter X. The author has found
it feasible to complete these ten chapters (and sometimes a bit more)
in the semester course which he has given along these lines at the Uni­
versity of Wisconsin.
The remaining five chapters can be used to extend such a course to a
year course, or as collateral reading to accompany a seminar.
A semester course on recursive functions for students having some prior
acquaintance with mathematical logic, or under an instructor with such
acquaintance, could start with Part III (Chapter IX). There are other
possibilities for selecting material; e.g. much of Part IV can follow directly
Part II or even Chapter VII for students primarily interested in mathe­
matical logic.
The author is indebted to Saunders MacLane for encouraging him to
write this book and for valuable criticism of an early draft of several
chapters. John Addison read the entire first printer’s proof with great care,
independently of the author. Among many others who have been of as­
sistance are Evert Beth, Robert Breusch, Arend Heyting, Nancy Kleene,
Leonard Linski, David Nelson, James Renno and Gene Rose. Scientific
indebtedness is acknowledged by references to the Bibliography; especially
extensive use has been made of Hilbert and Bemays’ "Grundlagen der
Mathematik” in two volumes and .
1934 1939
July 1952 S. C. Kleene
Note to the Sixth Reprint ( ). In successive reprints various errors have been
1971
corrected, the principal corrections being those listed in Jour, symbolic logic vol. 19
( ) p. 216 and vol. 33 ( ) pp. 290-291, and: on p. 505 bottom paragraph
1954 1968
sg((r)0) 'p{(r)-d + Wo '^(Wi) replaced by a function x{P> r) defined by Theorem
XX (c); on p. 506 allowance made in the middle paragraph for x possibly occurring
free in t, and line 5 from below " = ” changed to Moreover, in this sixth
reprint eleven bibliographical references have been updated (cf. end p. 517) and
two short notes have been added (on pp. 65 and 316).
TABLE OF CONTENTS
Part I. THE PROBLEM OF FOUNDATIONS
Chapter I. THE THEORY OF SETS........................................... 3
§ 1- Enumerable sets ................................................. 3
§ . Cantor's diagonal method..............................................
2 6
§ 3. Cardinal number.............................................................. 9
*§ 4. The equivalence theorem, finite and infinite sets . . .
11
*§ 5. Higher transfinite cardinals........................................... 14
Chapter II. SOME FUNDAMENTAL CONCEPTS................... 19
§ 6. The natural numbers...................................................... 19
§ 7. Mathematical induction..................................................
21
§ . Systems of objects.......................................................... 24
8
*§ 9. Number theory vs. analysis........................................... 29
§ . Functions................................................................. 32
10
Chapter III. A CRITIQUE OF MATHEMATICAL REASONING 36
The paradoxes .................................................................. 36
§ 11.
§ . First inferences from the paradoxes........................... 40
12
§ 13. Intuitionism...................................................................... 46
§ 14. Formalism......................................................................... 53
§ 15. Formalization of a theory.............................................. 59
Part II. MATHEMATICAL LOGIC
Chapter IV. A FORMAL SYSTEM.............................................. 69
§ 16. Formal symbols.............................................................. 69
§ 17. Formation rules .............................................................. 72
§ 18- Free and bound variables..................................... 76
§ 19. Transformation rules...................................................... 80
VIII TABLE OF CONTENTS
Chapter V. FORMAL DEDUCTION...........................................
86
§ . Formal deduction..............................
20 86
§ . The deduction theorem ................................................... 90
21
§ . The deduction theorem (concluded)............................... 94
22
§23. Introduction and elimination of logical symbols . . . 98
*§24. Dependence and variation.............................................
102
Chapter VI. THE PROPOSITIONAL CALCULUS.................... 108
§25. Proposition letter formulas................... 108
§26. Equivalence, replacement............................................... 113
§27. Equivalences, duality ...................................................... 118
§28. Valuation, consistency ................................................... 124
§29. Completeness, normal form ........................................... 130
§30. Decision procedure, interpretation............................... 136
Chapter VII. THE PREDICATE CALCULUS........................... 142
§31. Predicate letter formulas............................ . . . . 142
§32. Derived rules, free variables. .................................. . 146
§33. Replacement..................................................................... 151
Substitution...................................................................... 155
*§ 34.
§35. Equivalences, duality, prenex fo rm ...........................
162
§36. Valuation, consistency.................................................. 168
*§37. Set-theoretic predicate logic, ^-transforms . . . . . . 174
Chapter VIII. FORMAL NUMBER THEORY........................... 181
§38. Induction, equality, replacement.................................. 181
§39. Addition, multiplication, order. ............................... . 185
*§40. The further development oh number theory................ 189
Formal calculation.......................................................... 194
§41.
§42. GodeFs theorem.............................................................. 204
III. RECURSIVE FUNCTIONS
Part
Chapter IX. PRIMITIVE RECURSIVE FUNCTIONS . . . . 217
Primitive recursive functions.......................................
§43. 217
Explicit definition..........................................................
§44. 220
Predicates, prime factor representation.......................
§45. 223
TABLE OF CONTENTS IX
§46. Course-of-values recursion............................................. 231
*§47. Uniformity . . . . . . . . . . . . . . . . . . . 233
§48. Godel's p-function......................................................... 238
§49. Primitive recursive functions and the number-theoretic
formalism.......................................................................... 241
Chapter X. THE ARITHMETIZATION OF METAMATHE­
MATICS .......................................................................... 246
§50. Metamathematics as a generalized arithmetic . . . . 246
§51. Recursive metamathematical definitions.................... 251
§52. Godel numbering.............................................................. 254
*§ 53. Inductive and recursive definitions............................... 258
Chapter XI. GENERAL RECURSIVE FUNCTIONS................ 262
§54. Formal calculation of primitive recursive functions 262
§55. General recursive functions........................................... 270
§56. Arithmetization of the formalism of recursive functions 276
§57. The (i-operator, enumeration, diagonal procedure . . 279
§58. Normal form, Post’s theorem....................................... 288
*§ 59. General recursive functions and the number-theoretic
formalism.......................................................................... 295
§60. Church's theorem, the generalized Godel theorem. . . 298
§61. A symmetric form of Godel's theorem....................... 308
Chapter XII. PARTIAL RECURSIVE FUNCTIONS................ 317
§62. Church's th esis.............................................................. 317
§63. Partial recursive functions........................................... 323
§64. The 3-valued logic.......................................................... 332
§65. Godel numbers................................................................. 340
§ . The recursion theorem.................................................. 348
66
Chapter XIII. COMPUTABLE FUNCTIONS............................... 356
§67. Turing machines.............................................................. 356
§ . Computability of recursive functions........................... 363
68
§69. Recursiveness of computable functions.......................... 373
§70. Turing’s thesis.................................................................. 376
*§71. The word problem for semi-groups................................... 382
TABLE OF CONTENTS
X
Part IV. MATHEMATICAL LOGIC (ADDITIONAL TOPICS)
Chapter XIV. THE PREDICATE CALCULUS AND AXIOM
SYSTEMS......................................................................... 389
§72. Godel's completeness theorem ...................................... 389
§73. The predicate calculus with equality. . ........................ 399
*§ 74 Eliminability of descriptive definitions....................... 405
§75. Axiom systems, Skolem’s paradox, the natural number
sequence............................................................................ 420
§76. The decision problem . . . . . . . . . . . . . . . 432
Chapter XV. CONSISTENCY, CLASSICAL AND INTUITION-
ISTIC SYSTEMS.......................................................... 440
§77. Gentzen’s formal system .............................................. 440
§78. Gentzen's normal form theorem . ............................... 448
*§ 79. Consistency proofs.......................................................... 460
§80. Decision procedure, intuitionistic unprovability. . . . 479
§81. Reductions of classical to intuitionistic systems . . . 492
§82. Recursive realizability.............................................. . 501
BIBLIOGRAPHY.............................................................................. 517
SYMBOLS AND NOTATIONS....................................................... 538
INDEX................................................................................................. 539