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MATHEMATICAL LOGIC AND HILBERT'S [-SYMBOL A. C. LEISENRING Reed Col/ege, Portland, Oregon !J : i ! Ii , 1 MACDONALD TECHNICAL & SCIENTIFIC LONDON PREFACE MACDONALD & CO. (PUBLISHERS) LTD 49/50 POLAND STREET, LONDON W.1 This study of Hilbert's 8-symbol is based on a doctoral dissertation which was submitted to the University of London in June 1967. It contains a number of new results, in particular a strengthening of Hilbert's Second 8-Theorem, which should be of some interest to specialists in the field of mathematical logic. However, this book is not written for the expert alone. We have tried to make it as self-contained as possible so that most parts will be intelligible to any­ © 1969 A. C. Leisenring one with a good mathematical background but with a limited knowledge of SBN 35602679 5 formal logic. Since the 8-symbol can be used to overcome many technical diffi­ culties which arise in formalizing logical reasoning, the present book may be useful as supplementary reading material for an undergraduate course in logic. In Chapter I we define the formal languages which are used throughout the book and analyze the semantics of such languages. An interesting result concerning finitary closure operations is then established. This result is used in Chapter II to prove the (semantic) completeness of certain formal systems which incorporate the e-symbol. In Chapter II we also establish a number of derived rules of inference for these formal systems. Chapter III, which is more All Rights Reserved. No part of this publication may be reproduced, stored technical than Chapter II, contains proofs of Hilbert's 8-Theorems, Skolem's in a retrieval system, or transmitted, in any form or by any meanS, electronic, Theorem, and Herbrand's Theorem. If the reader is interested in only these mechanical, photocopying, recording or otherwise, without the prior permission of MacDonald & Co. (Publishers) Ltd. results, he may omit all of Chapter I except for the definitions of formal languages and may skim through most of Chapter II. Chapter IV deals with formal theories. We see how the e-symbol and c-Theorems may be used to prove the consistency of various mathematical theories. In particular, Hilbert's attempts to prove the consistency of arith­ metic are described. We then discuss the role which the 8-symbol can play in formalizing set theory, and we give particular attention to the relationship between this symbol and the axiom of choice. Chapter V may be regarded as an appendix to the book. In this chapter we give alternative proofs of some of the theorems in Chapter III. Our methods illustrate the close connection which exists between Hilbert's 8-Theorems and 'Gentzen's Hauptsatz. The author would like to express his gratitude to G. T. Kneebone (Bedford College, London) under whose helpful supervision his thesis was written. He is also indebted to Marian Cowan for her excellent work in typing the manu­ script, and most of all to his wife, Flora, for her help, patience, and en- couragement. A. C. Leisenring PRINTED AND nOUND IN ENGLAND BY Reed College HAZELL WATSON ANO VINEY LTD AYLESBURY, BUCKS Portland, Oregon CONTENTS Preface v Introduction I Objectives I 2 Formal languages 2 2.1 The semantic consequence relation F 2 2.2 The deducibility relation" 3 2.3 Formal theories and the formalist programme 3 3 The history of the e-symbol 4 4 The indeterminacy of the e-symbol 5 5 Possible ways of defining formal languages 6 Chapter I Syntax and Semantics I Introduction 9 2 The formal language 2'(11') 10 2.1 Terms and formulae 12 2.2 A unifying classification of formulae 14 2.3 The cardinality ofa language 16 3.1 Truth functions 16 3.2 Tautologies 17 3.3 Models 18 3.4 Satisfiability and the semantic consequence relation 22 4 The Satisfiability Theorem 23 4.1 Logical closures 30 5 Alternative interpretations of the e-symbol 32 6 Languages without identity 36 Chapter II Formal Systems 1.1 Finitary reasoning 37 1.2 General definitions 38 1.3 Axioms and rules of inference 39 2 The ,·calculus for 1/ 39 3 Derived rules of inference 41 3.1 The -+-introduction rule (Deduction Theorem) 42 3.2 Some theorems of the e-calculus 43 4 Completeness of the e-calculus 44 viii CONTENTS CONTENTS ix 5 The Tautology Theorem 45 4.4 The e-symbol and the axiom of choice 105 6 The consistency of the B-calculus 47 5.1 The predicate calculus with class operator and choice function 107 7.1 Some derived rules for operating with quantifiers 48 5.2 The relative consistency of set theory based on the B-calculus 112 7.2 Substitution instances and universal closures 51 7.3 The equivalence rule 52 Chapter V The Cut Elimination Theorem 7.4 Rule of relabelling bound variables 53 1 The sequent calculus 114 8 Prenex formulae 54 2 The axioms and rules of inference of the sequent calculus 115 9 The addition of new function symbols 54 2.1 Refutations 116 9.1 Skolem and Herbrand resolutions of prenex formulae 56 3 The subformula property of normal refutations 117 JO The elementary calculus 58 3.1 The eliminability of the identity symbol 118 II The predicate calculus 59 4 The contradiction rule lI8 ILl Derived rules of inference for the predicate calculus 61 4.1 The relationship between the sequent calculus and B-calculus 120 11.2 The predicate calculus without identity 62 5 k,Fk-refutations 121 J2 The formaJ superiority of the ,-calculus 63 5.1 The invertibility of the logical rules 121 6 The Cut Elimination Theorem 124 Chapter m The B-Theorems 6.1 Applications 126 I Introduction 64 2.1 The basic problem 65 Bibliography 129 2.2 Proper and improper formulae 66 2.3 The B'-calculus 66 Index of Symbols 134 2.4 The usefulness of the B'-calculus 68 3. J Subordination and rank 69 Index of Rules of Inference 137 3.2 r,Tr-deductions 7J 3.3 The Rank Reduction Theorem 72 General Index 139 3.4 The E2 Elimination Theorem 75 3.5 The First and Second e-Theorems 77 4.1 Skolem's Theorem 79 4.2 Herbrand's Theorem 80 4.3 The eliminability of the identity symbol 83 Chapter IV Formal Theories I Introduction 85 2 Finitary consistency proofs 85 3 Formal arithmetic 88 3.1 Numerals and numerically true formulae 90 3.2 Formal arithmetic based on the B-calculus 92 3.3 Quantifier-free proofs 94 3.4 The consistency of arithmetic 96 4 Theories based on the e-calculus 100 4.1 The I-symbol 100 4.2 Skolem functions 102 4.3 Definability of indeterminate concepts 103 INTRODUCTION 1 Objectives The purpose of this book is to examine the nature of Hilbert's e-symbol and to demonstrate the useful role which this logical symbol can play both in proving many of the classical theorems of mathematical logic and also in simplifying the formulation of logical systems and mathematical theories. The e-symbol is a logical constant which can be used in the formal languages of mathematical logic to form certain expressions known as e-terms. Thus, if A is a formula of some formal language 2 and x is a variable of 2, then the expression exA is a well-formed term of the language. Intuitively, the e-term exA says 'an x such that if anything has the property A, then x has that property'. For example, suppose we think of the variables of the language as ranging over the set of human beings and we think of A as being the state­ ment 'x is an honest politician'. Then 8xA designates some politician whose honesty is beyond reproach, assuming of course that such a politician exists. On the other hand, if there are no honest politicians, then exA must denote someone, but we have no way of knowing who that person is. Similarly, even if there are honest politicians, we have no way of knowing which one of them exA designates. Since the e-symbol, or e-operator as it is sometimes called, selects an arbitrary member from a set of objects having some given property, this symbol is often referred to as a 'logical choice function'. It is not surprising then that an investigation of the e-symbol also sheds some light on the nature of the axiom of choice. One of the main theorems of this book, Hilbert's Second e-Theorem, provides a formal justification of the use of the e-symbol in logical systems by showing that this symbol can be eliminated from proofs of formulae which do not themselves contain the symbol. What this theorem says intuitively is that the act of making arbitrary choices is a legitimate logical procedure. However, it has been shown by Cohen [1966J that an application of the axiom of choice in set theory cannot in general be elimin­ ated. It follows then that the real power of the axiom of choice lies not in the fact that it allows one to make arbitrary selections but rather in thc fact that it asserts the existence of a sct consisting of the selected entities. Before saying anything more about the nature of the e-symbol and the role it plays, we should first give a brief explanation of the basic objectives and concepts of mathematical logic. 2 INTRODUCTION §2.1 THE SEMANTIC CONSEQUENCE RELATION 3 2 Formal languages is the language of elementary group theory and X is the set of formal axioms The objects of study in mathematical logic are formal or symbolic lang­ of this theory.
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