An introduction to Gödel's theorems

406 Pages·2013·1.94 MB·English

by  Gödel

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
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Cambridge University Press
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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