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An Introduction to G¨Odel’Stheorems An Introduction to G¨odel’sTheorems Peter Smith Faculty of Philosophy University of Cambridge Version date: December 12, 2005 Copyright: c 2005 Peter Smith Not to be cited or quoted without permission The book’s website is at www.godelbook.net Contents Preface v 1 What G¨odel’s Theorems say 1 1.1 Basic arithmetic 1 1.2 Incompleteness 3 1.3 More incompleteness 4 1.4 Some implications? 5 1.5 The unprovability of consistency 6 1.6 More implications? 7 1.7 What’s next? 7 2 Decidability and enumerability 8 2.1 Effective computability, effective decidability 8 2.2 Enumerable sets 11 2.3 Effective enumerability 14 3 Axiomatized formal theories 15 3.1 Formalization as an ideal 15 3.2 Formalized languages 17 3.3 Axiomatized formal theories 20 3.4 More definitions 21 3.5 The effective enumerability of theorems 23 3.6 Negation complete theories are decidable 25 4 Capturing numerical properties 26 4.1 Three remarks on notation 26 4.2 A remark about extensionality 27 4.3 The language LA 28 4.4 Expressing numerical properties and relations 31 4.5 Capturing numerical properties and relations 33 4.6 Expressing vs. capturing: keeping the distinction clear 33 5 Sufficiently strong arithmetics 35 5.1 The idea of a ‘sufficiently strong’ theory 35 5.2 An undecidability theorem 36 5.3 An incompleteness theorem 37 5.4 The truths of arithmetic can’t be axiomatized 38 i Contents Interlude: taking stock, looking ahead 40 6 Two formalized arithmetics 43 6.1 BA – Baby Arithmetic 43 6.2 BA is complete 45 6.3 Q – Robinson Arithmetic 47 6.4 Q is not complete 48 6.5 Why Q is interesting 49 7 What Q can prove 50 7.1 Capturing less-than-or-equal-to in Q 50 7.2 Eight simple facts about what Q can prove 52 7.3 Defining the ∆0,Σ1 and Π1 wffs 54 7.4 Q is Σ1 complete 56 7.5 An intriguing corollary 58 8 First-order Peano Arithmetic 60 8.1 Induction and the Induction Schema 60 8.2 PA – First-order Peano Arithmetic 62 8.3 PA in summary 64 8.4 A very brief aside: Presburger Arithmetic 65 8.5 Is PA consistent? 66 9 Primitive recursive functions 69 9.1 Introducing the primitive recursive functions 69 9.2 Defining the p.r. functions more carefully 71 9.3 An aside about extensionality 73 9.4 The p.r. functions are computable 74 9.5 Not all computable numerical functions are p.r. 76 9.6 Defining p.r. properties and relations 78 9.7 Some more examples 79 10 Capturing functions 85 10.1 Expressing and capturing functions 85 10.2 ‘Capturing as a function’ 86 10.3 ‘If capturable, then capturable as a function’ 87 10.4 Capturing functions, capturing properties 88 10.5 The idea of p.r. adequacy 89 11 Q is p.r. adequate 91 11.1 Q can capture all Σ1 functions 91 11.2 LA can express all p.r. functions: starting the proof 94 11.3 The idea of a β-function 95 11.4 LA can express all p.r. functions: finishing the proof 97 11.5 The p.r. functions are Σ1 99 ii Contents 11.6 The adequacy theorem 100 Interlude: a very little about Principia 102 12 The arithmetization of syntax 108 12.1 G¨odel numbering 108 12.2 Coding sequences 110 12.3 Prfseq is p.r. 111 12.4 Some cute notation 112 12.5 The idea of diagonalization 113 12.6 Gdl and diag and are p.r. 113 12.7 Proving that Prfseq is p.r. 114 13 PA is incomplete 121 13.1 Constructing G 121 13.2 Interpreting G 122 13.3 G is undecidable in PA: the semantic argument 123 13.4 ‘G is of Goldbach type’ 124 13.5 G is unprovable in PA: the syntactic argument 124 13.6 ω-incompleteness, ω-inconsistency 125 13.7 ¬G is unprovable in PA: the syntactic argument 126 13.8 Putting things together 127 14 G¨odel’s First Theorem 128 14.1 Generalizing the semantic argument 128 14.2 Incompletability – a first look 130 14.3 The First Theorem, at last 130 14.4 Rosser’s improvement 132 14.5 Broadening the scope of the First Theorem 135 14.6 True Basic Arithmetic can’t be axiomatized 136 14.7 Incompletability – another quick look 137 15 Using the Diagonalization Lemma 138 15.1 The provability predicate 138 15.2 Diagonalization again 139 15.3 The Diagonalization Lemma: a special case 140 15.4 The Diagonalization Lemma generalized 141 15.5 Incompleteness again 142 15.6 Capturing provability? 142 15.7 Tarski’s Theorem 143 Interlude: about the First Theorem 147 16 The Second Incompleteness Theorem 151 16.1 Expressing the Incompleteness Theorem in PA 151 iii Contents 16.2 The Formalized First Theorem in PA 152 16.3 The Second Theorem for PA 153 16.4 How surprising is the Second Theorem? 154 16.5 How interesting is the Second Theorem? 156 17 Exploring the Second Theorem 158 17.1 More notation 158 17.2 The Hilbert-Bernays-L¨obderivability conditions 159 17.3 G, Con, and ‘G¨odelsentences’ 161 17.4 L¨ob’s Theorem 162 Bibliography 165 iv Preface In 1931, the young Kurt G¨odelpublished his First and Second Incompleteness Theorems; very often, these are simply referred to as ‘G¨odel’s Theorems’. His startling results settled (or at least, seemed to settle) some of the crucial ques- tions of the day concerning the foundations of mathematics. They remain of the greatest significance for the philosophy of mathematics – though just what that significance is continues to be debated. It has also frequently been claimed that G¨odel’sTheorems have a much wider impact on very general issues about language, truth and the mind. This book gives outline proofs of the Theorems and related formal results, and touches on some of their implications. Who is this book for? Roughly speaking, for those who want a lot more detail than you get in books for a general audience (the best of those is Franz´en 2005), but who find classic texts in mathematical logic (like Mendelson 1997) daunting and too short on explanatory scene-setting. So I hope philosophy students taking an advanced logic course will find the book useful, as will mathematicians who want a relatively relaxed exposition. I originally intended to write a rather shorter book, leaving more of the formal details to be filled in from elsewhere. But while that plan might have suited some readers, I soon realized that it would seriously irritate others to be sent hither and thither to consult a variety of text books with different terminologies and different notations. So in the end, I have given more or less full proofs of most key results. However, my original plan shows through in two ways. First, some proofs are still only roughly sketched in, and there are other proofs which are omitted entirely. Second, I try to signal very clearly when the detailed proofs I do give can be skipped without much loss of understanding. With judicious skimming, you should be able to follow the main formal themes of the book even if you start from a very modest background in logic.1 As we go through, there is also an amount of broadly philosophical commen- tary. I follow G¨odelin believing that our formal investigations and our general reflections on foundational matters should illuminate and guide each other. I hope that the more philosophical discussions (though certainly not always un- contentious) will also be reasonably widely accessible. Writing a book like this presents many problems of organization. At various points we will need to call upon some background ideas from general logical theory. Do we explain them all at once, up front? Or do we introduce them as 1I plan, in due course, to put optional exercises (and answers!) on the book’s website at www.godelbook.net. v Preface we go along, when needed? Similarly we will also need to call upon some ideas from the general theory of computation – for example, we will make use of both the notion of a ‘primitive recursive function’ and the more general notion of a ‘recursive function’. Again, do we explain these together? Or do we give the explanations many chapters apart, when the respective notions first get used? I’ve mostly adopted the second policy, introducing new ideas as and when needed. This has its costs, but I think that there is a major compensating benefit, namely that the way the book is organized makes it clearer just what depends on what. It also reflects something of the historical order in which ideas emerged. Many thanks are due to generations of students and to JC Beall, Hubie Chen, Torkel Franz´en, Andy Fugard, Jeffrey Ketland, Jonathan Kleid, Fritz Mueller, Tristan Mills, Jeff Nye, Alex Paseau, Michael Potter, Jos´eF. Ruiz, Wolfgang Schwartz and Brock Sides for comments on draft chapters. Particular thanks to Richard Zach both for pointing out a number of mistakes, large and small, in an early draft and for suggestions that much improved the book. And particular thanks too to Seamus Holland, Luca Incurvati, Brian King, and Mary Leng, who read carefully through a late draft in a seminar together and made many helpful comments. Like so many others, I am also hugely grateful to Donald Knuth, Leslie Lam- port and the LATEX community for the document processing tools which make typesetting a mathematical text like this one such a painless business.
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