An introduction to Gödel's theorems PDF
Preview An introduction to Gödel's theorems
more information – www.cambridge.org/9781107022843 An Introduction to Gödel’s Theorems In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book – extensively rewritten for its second edition – will be accessible to philosophy studentswithalimitedformalbackground.Itisequallysuitableformathematics students taking a first course in mathematical logic. Peter Smith was formerly Senior Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is also a former editor of the journal Analysis. An Introduction to Gödel’s Theorems Second edition Peter Smith University of Cambridge cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107022843 (cid:2)c Peter Smith 2007, 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2007 Second edition 2013 Printed and bound in the United Kingdom by the MPG Books Group Typeset by the author in Computer Modern 10/12.5 pt: system LATEX2ε A catalogue record for this publication is available from the British Library ISBN 978-1-107-02284-3 hardback ISBN 978-1-107-60675-3 paperback Additional resources for this publication at www.godelbook.net Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. For Patsy, as ever Contents Preface xiii Thanks xv 1 What Gödel’s Theorems say 1 Basic arithmetic · Incompleteness · More incompleteness · Some implica- tions? · The unprovability of consistency · More implications? · What’s next? 2 Functions and enumerations 8 Kinds of function · Characteristic functions · Enumerable sets · Enumer- ating pairs of numbers · An indenumerable set: Cantor’s theorem 3 Effective computability 14 Effectivelycomputablefunctions·Effectivelydecidablepropertiesandsets · Effective enumerability · Another way of defining e.e. sets of numbers · The Basic Theorem about e.e. sets 4 Effectively axiomatized theories 25 Formalization as an ideal · Formalized languages · Formalized theories · More definitions · The effective enumerability of theorems · Negation- complete theories are decidable 5 Capturing numerical properties 36 Three remarks on notation · The language LA · A quick remark about truth · Expressing numerical properties and functions · Capturing nu- merical properties and functions · Expressing vs. capturing: keeping the distinction clear 6 The truths of arithmetic 46 Sufficiently expressive languages · The truths of a sufficiently expressive language · Unaxiomatizability · An incompleteness theorem 7 Sufficiently strong arithmetics 49 The idea of a ‘sufficiently strong’ theory · An undecidability theorem · Another incompleteness theorem vii Contents 8 Interlude: Taking stock 53 Comparing incompleteness arguments · A road-map 9 Induction 56 Thebasicprinciple·Anotherversionoftheinductionprinciple·Induction and relations · Rule, schema, or axiom? 10 Two formalized arithmetics 62 BA,BabyArithmetic·BAisnegation-complete·Q,RobinsonArithmetic · Which logic? · Q is not complete · Why Q is interesting 11 What Q can prove 71 Capturing less-than-or-equal-to in Q · ‘≤’ and bounded quantifiers · Q is order-adequate · QcancorrectlydecideallΔ0 sentences · Σ1 andΠ1 wffs · Q is Σ1-complete · Intriguing corollaries · Proving Q is order-adequate 12 IΔ0, an arithmetic with induction 83 The formal Induction Schema · Introducing IΔ0 · What IΔ0 can prove · IΔ0 is not complete · On to IΣ1 13 First-order Peano Arithmetic 90 Being generous with induction · Summary overview of PA · Hoping for completeness · Is PA consistent? 14 Primitive recursive functions 97 Introducing the primitive recursive functions · Defining the p.r. functions more carefully · An aside about extensionality · The p.r. functions are computable · Not all computable numerical functions are p.r. · Defining p.r.propertiesandrelations · Buildingmorep.r.functionsandrelations · Further examples 15 L can express every p.r. function 113 A Starting the proof · The idea of a β-function · Finishing the proof · The p.r. functions and relations are Σ1-expressible 16 Capturing functions 119 Capturing defined · ‘Weak’ capturing · ‘Strong’ capturing 17 Q is p.r. adequate 124 The idea of p.r. adequacy · Starting the proof · Completing the proof · All p.r. functions can be captured in Q by Σ1 wffs 18 Interlude: A very little about Principia 130 Principia’s logicism · Gödel’s impact · Another road-map viii
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