0521857848 cambridge university press an introduction to gödels theorems aug 2007

An Introduction to Găodels Theorems In 1931, the young Kurt Gă odel 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 Gă odel also outlined an equally signicant Second Incompleteness Theorem How are these Theorems established, and why 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 will be accessible to philosophy students with a limited formal background It is equally suitable for mathematics students taking a first course in mathematical logic Peter Smith is Lecturer in Philosophy at the University of Cambridge His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal of Analysis An Introduction to Găodels Theorems Peter Smith University of Cambridge CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo 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/9780521857840 © Peter Smith 2007 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2007 eBook (MyiLibrary) ISBN-13 978-0-511-35096-2 ISBN-10 0-511-35096-1 eBook (MyiLibrary) ISBN-13 ISBN-10 hardback 978-0-521-85784-0 hardback 0-521-85784-8 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 What Găodels Theorems say xiii Basic arithmetic · Incompleteness · More incompleteness · Some implications? · The unprovability of consistency · More implications? · What’s next? Decidability and enumerability Functions · Effective decidability, effective computability · Enumerable sets · Effective enumerability · Effectively enumerating pairs of numbers Axiomatized formal theories Formalization as an ideal · Formalized languages · Axiomatized formal theories · More definitions · The effective enumerability of theorems · 17 Negation-complete theories are decidable Capturing numerical properties Three remarks on notation · A remark about extensionality · The language LA · A quick remark about truth · Expressing numerical properties and relations · Capturing numerical properties and relations · Expressing vs 28 capturing: keeping the distinction clear The truths of arithmetic 37 Sufficiently expressive languages · More about effectively enumerable sets · The truths of arithmetic are not effectively enumerable · Incompleteness Sufficiently strong arithmetics 43 The idea of a ‘sufficiently strong’ theory · An undecidability theorem · Another incompleteness theorem Interlude: Taking stock 47 Comparing incompleteness arguments · A road-map Two formalized arithmetics BA, Baby Arithmetic · BA is negation complete · Q, Robinson Arithmetic · Q is not complete · Why Q is interesting 51 vii Contents What Q can prove Systems of logic · Capturing less-than-or-equal-to in Q · Adding ‘≤’ to Q · Q is order-adequate · Defining the Δ0 , Σ1 and Π1 wffs · Some easy results · Q is Σ1 -complete · Intriguing corollaries · Proving Q is order-adequate 58 10 First-order Peano Arithmetic 71 Induction and the Induction Schema · Induction and relations · Arguing using induction · Being more generous with induction · Summary overview of PA · Hoping for completeness? · Where we’ve got to · Is PA consistent? 11 Primitive recursive functions 83 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 properties and relations · Building more p.r functions and relations · Further examples 12 Capturing p.r functions Capturing a function · Two more ways of capturing a function · Relating our definitions · The idea of p.r adequacy 99 13 Q is p.r adequate More definitions · Q can capture all Σ1 functions · LA can express all p.r functions: starting the proof · The idea of a β-function · LA can express all p.r functions: finishing the proof · The p.r functions are Σ1 · The adequacy theorem · Canonically capturing 106 14 Interlude: A very little about Principia Principias logicism ã Gă odels impact ã Another road-map 118 15 The arithmetization of syntax Gă odel numbering · Coding sequences · Term, Atom, Wff, Sent and Prf are p.r · Some cute notation · The idea of diagonalization · The concatenation function · Proving that Term is p.r · Proving that Atom and Wff are p.r · Proving Prf is p.r 124 16 PA is incomplete Reminders · ‘G is true if and only if it is unprovable’ · PA is incomplete: the semantic argument · ‘G is of Goldbach type’ · Starting the syntactic argument for incompleteness · ω-incompleteness, ω-inconsistency · Finishing the syntactic argument ã Gă odel sentences and what they say 138 17 Gă odels First Theorem 147 viii Contents Generalizing the semantic argument · Incompletability · Generalizing the syntactic argument · The First Theorem 18 Interlude: About the First Theorem What weve proved ã The reach of Gă odelian incompleteness ã Some ways to argue that GT is true · What doesn’t follow from incompleteness · What 153 does follow from incompleteness? 19 Strengthening the First Theorem 162 Broadening the scope of the incompleteness theorems · True Basic Arithmetic can’t be axiomatized · Rosser’s improvement · 1-consistency and Σ1 -soundness 20 The Diagonalization Lemma Provability predicates · An easy theorem about provability predicates · G and Prov · Proving that G is equivalent to ¬Prov( G ) · Deriving the 169 Lemma 21 Using the Diagonalization Lemma The First Theorem again ã An aside: Gă odel sentences again ã The Gă odelRosser Theorem again · Capturing provability? · Tarski’s Theorem · The Master Argument · The length of proofs 175 22 Second-order arithmetics 186 Second-order arithmetical languages · The Induction Axiom · Neat arithmetics · Introducing PA2 · Categoricity · Incompleteness and categoricity · Another arithmetic · Speed-up again 23 Interlude: Incompleteness and Isaacson’s conjecture Taking stock · Goodstein’s Theorem · Isaacson’s conjecture · Ever upwards · Ancestral arithmetic 199 24 Gă odels Second Theorem for PA Dening Con ã The Formalized First Theorem in PA · The Second Theorem for PA · On ω-incompleteness and ω-consistency again · How should we interpret the Second Theorem? · How interesting is the Second Theorem for PA? · Proving the consistency of PA 212 25 The derivability conditions More notation ã The Hilbert-Bernays-Lă ob derivability conditions ã G, Con, and Gă odel sentences · Incompletability and consistency extensions · The equivalence of fixed points for ¬Prov · Theories that ‘prove’ their own inconsistency ã Lă obs Theorem 222 ix ... at the University of Cambridge His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal of Analysis An Introduction to Găodels... Introduction to Găodels Theorems Peter Smith University of Cambridge CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh... something of the historical order in which ideas emerged My colleague Michael Potter has been an inspiring presence since I returned to Cambridge Many thanks are due to him and to all those who