THE CONTINUUM HYPOTHESIS: INDEPENDENCE AND TRUTH-VALUE BY THOMAS S. WESTON S.B., Massachusetts Institute of Technology 1966 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June, 1974 Signature of Author / Department of Philosophy, March 27, 1974 Certified by ._ Thesis Supervisor Accepted by Chairman, Departmental'Committee on Graduate Students L4 F~?A RIV-S ABSTRACT THE CONTINUUM HYPOTHESIS: INDEPENDENCE AND TRUTH-VALUE Submitted to the Department of Philosophy, M.I.T., in partial fulfillment of the requirements for the degree of Doctor of Philosophy. In the Introduction, Cantor's continuum hypothesis (CH) is stated, and the history of attempts to prove it is reviewed. The major problems of the thesis are stated: Does the CH have a truth-value, and if so, how could we discover what it is, given the proofs of its independence from the most widely accepts set theories? By way of a prima facie case that CH has no truth-value, several versions of formalist accounts of truth in mathematics are described. Chapter I examines the claim that the independence proofs for CH are only possible because of inadequacies in first-order formalized set theory. Various strategies for repairing or generalizing formal systems are examined and it is shown that no so far proposed strategy based on the usual axioms of set theory excapes independence proofs. The latter part of the chapter discusses G. Kreisel's claim that the CH is "decided" in second - order formulations of Zermelo-Fraenkel set theory. It is argued that (a) second-order logic is of no heuristic value in discovering a proof or refutation of CH, and that (b) Kreisel's semantic arguments that CH is "decided" are sound only if a much simpler argument that does not used second-order logic is also sound. This simpler argument is investigated in the second chapter. In Chapter II, the question of the truth-value of CH is examined from the point of view of various sorts of realism. A notion of "minimal realism" is formulated which forms the basis of subsequent discussion. It is argued that this form of realism is plausible, but that it cannot be sufficiently clarified at present to exclude the possibility that there may be alternative equally natural interpretations of set theory, some according to which the CH is true, and others according to which it is false. Various historical cases are examined to show that the possibility of multiple natural interpretations has been a major difficulty in other problems in mathematics. The model construction techniques of Godel and Cohen and their generalizations are examined to see if they provide equally plausible interpretations of set theory, and it is concluded that they do not. In the final section, it is argued that although we cannot conclusively rule out multiple interpretations of set theory, it is very implausible that the portion of "the universe" of sets which concerns the CH has multiple interpretations, and that the fact of independence results does not increase the plausibility of this view. In Appendix A, various types of plausibility arguments in mathematics are sketched, and the controversies over the plausibility of the CH and the axiom of choice are reviewed and compared. It is concluded that the axiom of choice should be regarded as well established, but no existing arguments show that CH or its negation are plausible. Three technical appendices prove results used in the text and summarize axiomatic theories discussed there. The author gratefully acknowledges the patient help of his committee, Professors Richard Cartwright and George Boolos. Invaluable advice and moral support were also provided by Professors Richard Boyd and Hillary Putnam. Thesis Committee: Professor Richard Cartwright, Chairman Professor George Boolos TABLE OF CONTENTS ABSTRACT 2 INTR ODUC TION 6 What is the Continuum Hypothesis ? 6 History of the Continuum Question 6 Present Status 7 CH is not determined: A prima facie case 9 CHAPTER I 12 Introduction 12 Fixing Up Formal Theories 16 Two Devices for "Completing"Arithmetic 19 Other ways to "fix" ZF 22 Non-Standard Models and Expressibility 25 "Unrestricted Concept of Predicate": Background 28 Restrictions on definite predicates and independence 33 Von Neumann's system 35 Natural Models for ZF, VBG & VBI 38 VBG and VBI Compared 41 Zermelo's Proposal 43 I"Superclass" Theories 45 Nodal Type Theory 47 NTT & CH 51 NTT & Informalism 52 Why Consider Second-Order Logic 53 Arithmetic: A Second-Order Example 55 Properties of SOL 59 I. S2 is Categorical 60 II. :*SOL is not Semantically Complete 60 General Structures & SOL 62 Second-Order Set Theory 62 Comparing VBI & ZF 2 63 A "Categoricity" Result for ZF 2 64 Significance of the "Categoricity" of *ZF 2 66 Claimed Advantages of *SOL 67 Evidential Superiority of *SOL 70 Page Heuristic Value of SOL 71 Further Disadvantages of *ZF2 74 A Better Argument that CH is Determined 75 The Critical "Assumption" 77 CHAPTER II: REALISM AND THE CONTINUUM QUESTION 78 Setting the Problem 78 Realist Positions in Mathematics 81 Quasi-R ealist Views 84 Evaluation of Realist & Quasi-Realist Views 87 Minimal Realism 88 Theories of Truth in Mathematics 89 Theory I: Conventionalism 89 Tarski's Views 92 Tar ski' s Theory of Primitive Denotation 94 Status of the Theory T 99 Reference, Fulfillment and Application in Model Theory 101 Set Theoretic Accounts of Primitive Denotation for 'e' 103 Application for ' ' in Other Set Theories 105 Further Attempts to Explain Application of ' ' 108 Cantor's Proper Classes 110 Proper Classes as Definite Properties (Again) 114 The Universe Described in English 116 Set Theory vs. Number Theory 120 Alternative Set Theories 122 Natural Interpretations of ZF 126 Historical Cases of Unclarity of Mathematical Notions 131 The Vibrating String in the 18th Century 133 Non-Euclidean Geometry 137 The Historical Cases Summed Up 140 The Independence Results and Multiple Interpretations 141 The Constructible Universe 143 Cohen' s Constructions 147 Undecided Questions: Do They Show Unclarity? 150 A Final Word on Independence and Multiple Interpreations 154 5 Argument (A) Reconsidered 157 Up to the Next Rank, R(w + 2) 163 Conclusion 164 APPENDIX A: PLAUSIBILITY ARGUMENTS 165 Introduction 165 Plausibility Arguments 165 Origin and Early Controversies about AC 169 Well-Ordering and the CH 170 Later Controversies about AC 172 Present Status of AC 174 AC and the Physical Sciences 175 "Intuitive Evidence" of AC 176 Alternatives to AC--Their Relative Plausibility 178 The axiom of determinateness 179 Plausibility of AD 182 Consequences of CH 187 Non-Measurable Sets of Reals 188 Relations of Measure and Category 189 Existence of "Sparse" sets 189 The "Implausibility" of (6) -(11) 190 Explaining Away Implausibility 193 Cardinality Questions and Plausibility 195 Hypotheses Incompatible with CH 196 Plausibility of CH Summed Up 198 Other Principles 198 APPENDIX B: AXIOM SYSTEMS USED IN THE TEXT 201 APPENDIX C: EQUIVALENCE OF VBI AND ZF 2 209 APPENDIX D: "CATEGORICITY' OF :"ZF 2 212 FOOTNO TES- -INTR ODUC TION 215 FOOTNOTES--CHAPTER I 217 FOOTNOTES- -CHAPTER II 227 FOOTNOTES- -APPENDIX A 248 FOOTNOTES--APPENDIX D 258 IN TR ODUC TION What is the Continuum Hypothesis ? (CH) "Every infinite set of real numbers can be put into one-to-one correspondence either with the set of natural numbers or with the set of all real numbers." This is, in essence, the continuum hypothesis, first proposed by Georg Cantor in 1878.1 In the presence of the axiom of choice, which will be assumed throughout most of this thesis, CH is equivalent to the Aleph Hypothesis: (AH) 2 = That is, the cardinal of the continuum is the first uncountable cardinal. Cantor himself accepted the AC, or rather its equivalent, the statement that every set can be well-ordered,2 so this distinction was unnecessary for him. In the sequel, we shall distinguish CH from AH only in Appendix A, where the status of AC is discussed. Since we will be discussing the CH in the context of set theory, we will always mean by 'the continuum' the set of all sets of finite ordinals, IP(w), to which the real numbers are isomorphic in a natural and familiar way. The History of the Continuum Question At the time when he proposed CH, Cantor promised that "an exact solution to this question" closed by his "theorem" would come later.3 There followed a long series of unsuccessful attempts to deliver on this promise, but none of his attempts at proof succeeded. 4 In 1904 and again in 1905, Konig claimed to have shown that CH is false, but these arguments were faulty and were withdrawn. 5 In 1900, Hilbert had listed the continuum problem first on his famous list of outstanding unsolved mathematical problems. 6 In 1925, he announced a proof of CH which also proved faulty. The only correct partial result from this collection of faulty proofs in Konig's "lemma" that 2 is not the limit of countably many smaller cardinals. 8 In 1938, Kurt Godel was able to prove that if Zermelo- Frankel set theory (ZF) 9 is consistent, then it remains so on addition of AC and CH. Godel accomplished this by showing that the sets satisfying the apparently restrictive condition of con- structibility10 form an interpretation satisfying ZF and AC and CH.11 The relative consistency of NCH with the axioms of ZF cannot be shown by an analogous method,12 however, and this result was only obtained 25 years later by Paul Cohen.13 His completion of the proof that CH is independent of the axioms of ZF, even if the axiom of choice is assumed, confirmed a conjecture of Skolem made in 1922.