Computability and logic

370 Pages·2002·1.769 MB·English

by  George S. Boolos, John P. Burgess, Richard C. Jeffrey

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ComputabilityandLogic,FourthEdition
ThisfourtheditionofoneoftheclassiclogictextbookshasbeenthoroughlyrevisedbyJohn
Burgess. The aim is to increase the pedagogical value of the book for the core audience
ofstudentsofphilosophyandforstudentsofmathematicsandcomputerscienceaswell.
Thisbookhasbecomeaclassicbecauseofitsaccessibilitytostudentswithoutamathe-
maticalbackground,andbecauseitcoversnotsimplythestapletopicsofanintermediate
logic course such as Go¨del’s incompleteness theorems, but also a large number of op-
tional topics from Turing’s theory of computability to Ramsey’s theorem. John Burgess
has enhanced the book by adding a selection of problems at the end of each chapter and
byreorganizingandrewritingchapterstomakethemmoreindependentofeachotherand
thustoincreasetherangeofoptionsavailabletoinstructorsastowhattocoverandwhatto
defer.
“...givesanexcellentcoverageofthefundamentaltheoreticalresultsaboutlogicinvolving
computability,undecidability,axiomatization,definability,incompleteness,etc.”American
MathematicalMonthly
“Thewritingstyleisexcellent:althoughmanyexplanationsareformal,theyareperfectly
clear.Modern,elegantproofshelpthereaderunderstandtheclassictheoremsandkeepthe
booktoareasonablelength.”ComputingReviews
“...avaluableassettothosewhowanttoenhancetheirknowledgeandstrengthentheirideas
intheareasofartificialintelligence,philosophy,theoryofcomputing,discretestructures,
mathematicallogic.Itisalsousefultoteachersforimprovingtheirteachingstyleinthese
subjects.”ComputerEngineering

Computability and Logic
Fourth Edition
GEORGE S. BOOLOS
JOHN P. BURGESS
PrincetonUniversity
RICHARD C. JEFFREY
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521809757
© George S. Boolos, John P. Burgess, Richard Jeffrey 2002
This book 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 2002
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For
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and
EDITH

Contents
Preface pagex
COMPUTABILITY THEORY
1 Enumerability 3
1.1 Enumerability 3
1.2 EnumerableSets 7
2 Diagonalization 16
3 TuringComputability 23
4 Uncomputability 35
4.1 TheHaltingProblem 35
4.2 TheProductivityFunction 40
5 AbacusComputability 45
5.1 AbacusMachines 45
5.2 SimulatingAbacusMachinesbyTuringMachines 51
5.3 TheScopeofAbacusComputability 57
6 RecursiveFunctions 63
6.1 PrimitiveRecursiveFunctions 63
6.2 Minimization 70
7 RecursiveSetsandRelations 73
7.1 RecursiveRelations 73
7.2 SemirecursiveRelations 80
7.3 FurtherExamples 83
8 EquivalentDefinitionsofComputability 88
8.1 CodingTuringComputations 88
8.2 UniversalTuringMachines 94
8.3 RecursivelyEnumerableSets 96
vii
viii CONTENTS
BASIC METALOGIC
9 APre´cisofFirst-OrderLogic:Syntax 101
9.1 First-OrderLogic 101
9.2 Syntax 106
10 APre´cisofFirst-OrderLogic:Semantics 114
10.1 Semantics 114
10.2 MetalogicalNotions 119
11 TheUndecidabilityofFirst-OrderLogic 126
11.1 LogicandTuringMachines 126
11.2 LogicandPrimitiveRecursiveFunctions 132
12 Models 137
12.1 TheSizeandNumberofModels 137
12.2 EquivalenceRelations 142
12.3 TheLo¨wenheim–SkolemandCompactnessTheorems 146
13 TheExistenceofModels 153
13.1 OutlineoftheProof 153
13.2 TheFirstStageoftheProof 156
13.3 TheSecondStageoftheProof 157
13.4 TheThirdStageoftheProof 160
13.5 NonenumerableLanguages 162
14 ProofsandCompleteness 166
14.1 SequentCalculus 166
14.2 SoundnessandCompleteness 174
14.3 OtherProofProceduresandHilbert’sThesis 179
15 Arithmetization 187
15.1 ArithmetizationofSyntax 187
15.2 Go¨delNumbers 192
15.3 MoreGo¨delNumbers 196
16 RepresentabilityofRecursiveFunctions 199
16.1 ArithmeticalDefinability 199
16.2 MinimalArithmeticandRepresentability 207
16.3 MathematicalInduction 212
16.4 RobinsonArithmetic 215
17 Indefinability,Undecidability,Incompleteness 221
17.1 TheDiagonalLemmaandtheLimitativeTheorems 221
17.2 UndecidableSentences 225
17.3 UndecidableSentenceswithouttheDiagonalLemma 227
18 TheUnprovabilityofConsistency 233