Graduate Texts in Mathematics 53 Editorial Board F W Gehring P.R Halmos Managing Editor C C Moore Yu I Manin A Course in Mathematical Logic Translated from the Russian by Neal Koblitz Springer Science+Business Media, LLC Neal Koblitz Yu I Manin Department of Mathematics Harvard University Cambridge, Massachusetts 02138 CSA V A Steklov Mathematical Institute of the Academy of Sciences Moscow V-333 UI Vavilova 42 USSR Editorial Board P.R Halmos F W Gehring C C Moore Managing Editor Department of Mathematics University of Michigan Ann Arbor, Michigan 48104 USA Department of Mathematics University of California at Berkeley Berkeley, California 94720 USA Department of Mathematics University of California Santa Barbara, California 93106 USA AMS Subject Classifications: 02-01, 02Bxx, 02Fxx Library of Congress Cataloging in Publication Data Manin, IV A course in mathematical logic (Graduate texts in mathematics; 53) Includes index Logic, Symbolic and mathematical II Series 77-1838 511'.3 QA9.M296 I Title All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC © 1977 by Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1977 ISBN 978-1-4757-4385-2 (eBook) ISBN 978-1-4757-4387-6 DOI 10.1007/978-1-4757-4385-2 To my son Preface This book is above all addressed to mathematicians It is intended to be a textbook of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last ten or fifteen years These include: the independence of the continuum hypothesis, the Diophantine nature of enumerable sets, the impossibility of finding an algorithmic solution for one or two old problems All the necessary preliminary material, including predicate logic and the fundamentals of recursive function theory, is presented systematically and with complete proofs We only assume that the reader is familiar with "naive" set theoretic arguments In this book mathematical logic is presented both as a part of mathematics and as the result of its self-perception Thus, the substance of the book consists of difficult proofs of subtle theorems, and the spirit of the book consists of attempts to explain what these theorems say about the mathematical way of thought Foundational problems are for the most part passed over in silence Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life The first two chapters are devoted to predicate logic The presentation here is fairly standard, except that semantics occupies a very dominant position, truth is introduced before deducibility, and models of speech in formal languages precede the systematic study of syntax The material in the last four sections of Chapter II is not completely traditional In the first place, we use Smullyan's method to prove Tarski's theorem on the undefinability of truth in arithmetic, long before the vii Preface introduction of recursive functions Later, in the seventh chapter, one of the proofs of the incompleteness theorem is based on Tarski's theorem In the second place, a large section is devoted to the logic of quantum mechanics and to a proof of von Neumann's theorem on the absence of "hidden variables" in the quantum mechanical picture of the world The first two chapters together may be considered as a short course in logic apart from the rest of the book Since the predicate logic has received the widest dissemination outside the realm of professional mathematics, the author has not resisted the temptation to pursue certain aspects of its relation to linguistics, psychology, and common sense This is all discussed in a series of digressions, which, unfortunately, too often end up trying to explain "the exact meaning of a proverb" (E Baratynskii 1) This series of digressions ends with the second chapter The third and fourth chapters are optional They are devoted to complete proofs of the theorems of Godel and Cohen on the independence of the continuum hypothesis Cohen forcing is presented in terms of Boolean-valued models; Godel's constructible sets are introduced as a subclass of von Neumann's universe The number of omitted formal deductions does not exceed the accepted norm; due respects are paid to syntactic difficulties This ends the first part of the book: "Provability." The reader may skip the third and fourth chapters, and proceed immediately to the fifth Here we present elements of the theory of recursive functions and enumerable sets, formulate Church's thesis, and discuss the notion of algorithmic undecidability The basic content of the sixth chapter is a recent result on the Diophantine nature of enumerable sets We then use this result to prove the existence of versal families, the existence of undecidable enumerable sets, and, in the seventh chapter, Godel's incompleteness theorem (as based on the definability of provability via an arithmetic formula) Although it is possible to disagree with this method of development, it has several advantages over earlier treatments In this version the main technical effort is concentrated on proving the basic fact that all enumerable sets are Diophantine, and not on the more specialized and weaker results concerning the set of recursive descriptions or the Godel numbers of proofs Nineteenth century Russian poet (translator's note) The full poem is: We diligently observe tlte world, We diligently observe people, And we hope to understand tlteir deepest meaning But what is tlte fruit of long years of study? What tlte sharp eyes finally detect? What does the haughty mind finally learn At the height of all experience and thought, What?-tlte exact meaning of an old proverb 1828 Vlll Preface The last section of the sixth chapter stands somewhat apart from the rest It contains an introduction to the Kolmogorov theory of complexity, which is of considerable general mathematical interest The fifth and sixth chapters are independent of the earlier chapters, and together make up a short course in recursive function theory They form the second part of the book: "Computability." The third part of the book, "Provability and Computability," relies heavily on the first and second parts It also consists of two chapters All of the seventh chapter is devoted to Godel's incompleteness theorem The theorem appears later in the text than is customary because of the belief that this central result can only be understood in its true light after a solid grounding both in formal mathematics and in the theory of computability Hurried expositions, where the proof that provability is definable is entirely omitted and the mathematical content of the theorem is reduced to some version of the ''liar paradox," can only create a distorted impression of this remarkable discovery The proof is considered from several points of view We pay special attention to properties which not depend on the choice of Godel numbering Separate sections are devoted to Feferman's recent theorem on Godel formulas as axioms, and to the old but very beautiful result of Godel on the length of proofs The eighth and final chapter is, in a way, removed from the theme of the book In it we prove Higman's theorem on groups defined by enumerable sets of generators and relations The study of recursive structures, especially in group theory, has attracted continual attention in recent years, and it seems worthwhile to give an example of a result which is remarkable for its beauty and completeness This book was written for very personal reasons After several years or decades of working in mathematics, there almost inevitably arises the need to stand back and look at this research from the side The study of logic is, to a certain extent, capable of fulfilling this need Formal mathematics has more than a slight touch of self-caricature Its structure parodies the most characteristic, if not the most important, features of our science The professional topologist or analyst experiences a strange feeling when he recognizes the familiar pattern glaring out at him in stark relief This book uses material arrived at through the efforts of many mathematicians Several of the results and methods have not appeared in monograph form; their sources are given in the text The author's point of view has formed under the influence of the ideas of Hilbert, Godel, Cohen, and especially John von Neumann, with his deep interest in the external world, his open-mindedness and spontaneity of thought Various parts of the manuscript have been discussed with Yu V Matijasevic, G V Cudnovskii, and S G Gindikin I am deeply grateful to all of these colleagues for their criticism lX Preface W D Goldfarb of Harvard University very kindly agreed to proofread the entire manuscript For his detailed corrections and laborious rewriting of part of Chapter IV, I owe a special debt of gratitude I wish to thank Neal Koblitz for his meticulous translation Yu I Manin Moscow, September 1974 Interdependence of Chapters I I I ~8 /1\/ X Contents Part I PROVABILITY I Introduction to formal languages I General information First order languages Digression: names Beginners' course in translation Digression: syntax II Truth and deducibility Unique reading lemma Interpretation: truth, definability Syntactic properties of truth Digression: natural logic Deducibility Digression: proof Tautologies and Boolean algebras Digression: Kennings Godel's completeness theorem Countable models and Skolem's paradox Language extensions Undefinability of truth: the language SELF 3 10 16 20 20 24 30 34 38 48 51 56 58 64 69 73 "A metaphorical compound word or phrase used especially in Old English and Old Norse poetry, e.g., 'swan-road' for 'ocean' "-Webster's New Collegiate Dictionary (translator's note) XI Bounded systems of generators Similarly, we have I; (E) u ;(G)I n; (G)= IE u Fl in G* FG The notation is compatible with the fact that these two intersections are identified in the amalgamation of the product M M , which is constructed as in part (a) Applying 2.3(d) to this product, we find that * li'(I I, be the group freely generated by the a; We call a subset R' c G' bounded if there exists an r > such that any element in R' can be represented in the form a;~' · · · a(', X; E Z In this section we prove the following special case of the hypothesis of Proposition 4.3: 5.2 Proposition If the subgroup H' c G' is generated by a bounded enumerable subset R' c G', then it is benign 273 VIII Recursive groups Corollary The same is true Lemma 4.5) if G' is an f.g subgroup of an f.p group (using In the next section we show how the general case follows from this special case The proof of 5.2 consists of a series of reduction steps 5.3 First reduction In the free group G = i{a 1, b 1, c1; shall consider a set of "layered" words of the form • • • ;arn, b,n, crn}l we and the subgroup H c G it generates We shall later show that, if R is enumerable, then H is benign This is a special case of 5.2 to which the general case reduces using the following technique Suppose we are given G' and R' as in 5.1 For each element g' = a/'I · · · a/' E R' we construct an element g E G as follows We represent ' g' in the form n n II at'·' II at i=l i=l n 2·' • II ax,,; i=l I ' where fori= ik, fori=!= ik We then set If R' is enumerable, then the set R of all elements g obtained from all the g' E R' is enumerable We consider the surjective homomorphism l { b;aJ b; , ~• < ~• 1;;; z a1, a;bA-I·~ < ~ { aJ a; -10a;, ~ ;; J;;; ap b) { a; -lb1 a;, < z j ;;;>I b), { { { b), j ;;;> I b;-tbA,~< ~ b), J ;;;>I { b;bA-~.~ < ~ b), J > l We finally set: K' = jG' U { x;, x;.Y;.Y;Ii;; 0}: the relations in the tablej, and we take G' ~ K' to be the natural embedding 281 VIII Recursive groups 6.5 The group L' We set L' =I{ tcde, X;, X;,Y;.Y;Ii ~ O}i C K', and we take L' ~ K' to be the natural embedding In subsection 6.7 we shall verify that H' is embedded in L' (as a subgroup of K', in view of the commutativity of the diagram) 6.6 The groups K and L We set K=IG' U {u 1, u2 , u3, u4 , v 1, v2, v3, v4 } : Rl, where the relations R and the embedding K' ~ K are both defined by the conditions R = the image of the relations in the table after making the substitutions K' ~K is the homomorphism which is the identity on G' and acts by these substitutions on the other generators The homomorphism K' ~K is an embedding In fact, the elements ui-;viu/ are free in I{ ui, vi} I, so that K can be considered as the free product of K' and I{~· 1111 < j ;;; 4} I with the amalgamation given by the above substitutions (here we take into account Proposition 2.7(a)) Finally, we set L = the image of L' under the embedding K' ~ K 6.7 The diagram has now been constructed It follows immediately from the definitions that it satisfies 6.4(a) and (b) It remains to show that H' = G' n L' inK' (a) We set [n] = -r(g)cnden for n E z+ and g = y(n) in the notation of 6.1 We recall that H' is generated by all the [n] in G', and hence inK' as well The table of relations in K' was composed in such a way so that the following relations would be fulfilled: X;- 1[n]x; = [p ;n], Y;- 1[n]y; = [P:i+2], x;- 1[n]x; = [P ;+ 1n], Y;- 1[n]y; = [P4i+Jn] For example, we verify the first relation Let n = IIptJ Then, according to the definitions, y(n) = ITam4;-m4J+1bm4J+2-m4;+3, j [ n] = t Ilajm4] -m4j+lbjm4j+2-m4J+>cnden, j 282 End of the proof so that, by the first column of the table in 6.4, X;- I [ n ]x; = ta;a;-l II ( · · · ); a; II ( · · · ); cP•,ndehn = j O}l = L, since the inclu- sion c has been verified, and the inclusion ::J is obvious (b) We now show that in K' we have JH' u {x;, x;,Y;.Y;Ii > o}Jn G' = H' Since K' is an HNN-extension of G', it suffices to show that we are in the situation of Proposition 2.7 (as described in the paragraph preceding the proposition, at the end of 2.6), and then to apply 2.7(b) We verify these conditions, for example, for the first series of isomorphisms of the subgroup of G', as described at the beginning of 6.4 This series corresponds to conjugating by X; in K' The conditions take the following form in our case: x;- [ H' n j{t, c, d, e; ai' b11J > O}J]x; = H' n X;-'J{t, c, d, e; aj, bjlj > o}Jx;; i.e., if we use the definition of H' and the table, x;- 1H'x; = H' n J{t, cP.,, d, eP4i; ai' b)j > O}J inclusion c is obvious Conversely, Since x;- 1[n]x; = (p4;n], the suppose we are given an element in H' which is written as a reduced word in the [n]: IIn, [n_;]'1, ey = ± We consider the corresponding reduced word g in G' We show that if all the powers of c and d which occur in g are divisible by p 4;, then all the n1 with nonzero ey in the above product are divisible by p 4;, i.e., [n1] E X;- 1H'x; In fact, let g =the image of g in I{ c, d, e }I under the homomorphism which takes t, a1, and b1 to Since [il] = cnden, it follows that all the [il] are free, and that g uniquely determines the sequence {e1n1 } It is not hard to see that the formulas which express e1 n1 in terms of the powers of c and e which occur in the reduced word g are linear with integer coefficients (more precisely, they are a disjunction of linear formulas accompanied by inequality c~nditions) Therefore, if all these powers are divisible by p 4; then so is n_; This completes the proof D 283 Index Absolute 65, 155 Admissible 93, 104, 235 Algorithmically decidable 183 Alphabet Atomic Axiom of Choice 47, 135 Axiom Schema 43 Benign (subgroup) 270 Big class 151 Boolean algebra 54 Boolean truth function 55, 120 Bounded 273 Cardinality 103 Characteristic function 179 Church's thesis 181 Closed 24 Codimension 191 Cofinality 148 Cohen forcing 55, 143ff Compatible (numberings) 234 Composition 181 Computable 178 Consistent 58 Constant Constructible 150, 167 Continuum hypothesis 105 Decidable 198 Diophantine 206 Disjoint 146 Display 79 Effective (numbering) Expression 152 Family 101 Finitely generated 262 Finitely presented 262 First order language Formula Free 23 Gen (generalization) 30 General recursive 183 Generalized continuum hypothesis 160 Generic 145 Godel numbering 242 Godel's completeness theorem 39, 58 Godel's incompleteness theorem 81, 239 Godelian (set of formulas) 32 Hereditarily finite 45 Higher order languages 15 Hilbert's tenth problem 208 Inaccessible cardinal 46 Inconsistent 58 Internal model 171 Interpretation 24 285 Index Juxtaposition 182 Kenning 56 Kolmogorov's complexity theorem 226 L 1Ar L 1Set ~Real 109 Label 79 Language Level set 190 Logical polynomial 30 Lowenheim-Skolem theorem JL-operator 182 Mapping 25 Mass problem 183 Metalanguage Model 26 Modular structure 92 MP (modus ponens) 30 Name Normal models 43 Numbering 233 Occurrence 20 Ordinal 96 Parentheses bijection 21 Partial Boolean algebra 89 Partial function 178 Partial ordering 96 Partial recursive 183 Primitive enumerable 191 Primitive recursive 183 Proof 48 Q-absolute 171 Quantum logic 87 Quantum tautology 91 Quasitopology 201 286 65 Random class 126 Reality Recursion 182 Recursive (group) 262 Recursively enumerable 190 Reduced (word) 261 Relativization (of a formula) 171 Representation 31 SAr 16, 74 SELF 16, 73 Semantics Semi-computable 178 Separable (ordering) 146 Set theory (Zermelo-Fraenkel) 44 Simple 205 Singular 148 Skolem's paradox 68 Special axioms (of L) 38 Standard interpretation 26 Standard model Syntax 4, 16 Tarski's theorem 79, 240 Tautology 31, 54 Term Text Total function 203 Transfinite recursion 99 Truth function 26 Undefinability theorem 74 V 121 Variables Versa) 203 Von Neumann theorem 82 Von Neumann universe 100 Well-ordering 96 Word identities (in groups) 184 Graduate Texts in Mathematics Soft and hard cover editions are available for each volume up to vol 14, hard cover only from Vol 15 10 11 12 13 TAKEUTI/ZARING Introduction to Axiomatic Set Theory vii, 250 pages 1971 OXTOBY Measure and Category viii, 95 pages 1971 ScHAEFFER Topological Vector Spaces xi, 294 pages 1971 HILTON/STAMMBACH A Course in Homological Algebra ix, 338 pages 1971 (Hard cover edition only) MACLANE Categories for the Working Mathematician ix, 262 pages 1972 HUGHEs/PIPER Projective Planes xii, 291 pages 1973 SERRE A Course in Arithmetic x, 115 pages 1973 TAKEUTI/ZARING Axiomatic Set Theory viii, 238 pages 1973 HUMPHREYS Introduction to Lie Algebras and Representation Theory xiv, 169 pages 1972 CoHEN A Course in Simple Homotopy Theory xii, 114 pages 1973 CoNwAY Functions of One Complex Variable 2nd corrected reprint xiii, 313 pages 1975 (Hard cover edition only.) 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Smullyan in his language of arithmetic; see § 10 of Chapter II 17 I Introduction to formal languages General remarks Most natural and artificial languages are characteristically discrete and linear
