Models and games

381 Pages·2011·2.39 MB·English

by  Vaananen J.

This page intentionally left blank
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 132
EditorialBoard
B.BOLLOBÁS, W.FULTON, A.KATOK,
F.KIRWAN, P.SARNAK, B.SIMON, B.TOTARO
ModelsandGames
Thisgentleintroductiontologicandmodeltheoryisbasedonasystematicuseofthree
importantgamesinlogic:theSemanticGame,theEhrenfeucht–FraïsséGame,andthe
ModelExistenceGame.Thethirdgamehasnotbeenisolatedintheliteraturebefore,
butitunderliestheconceptsofBethtableauxandconsistencyproperties.
JoukoVäänänenshowsthatthesegamesarecloselyrelatedand,inturn,governthe
threeinterrelatedconceptsoflogic:truth,elementaryequivalence,andproof.Allthree
methodsaredevelopednotonlyforfirst-orderlogic, butalsoforinfinitarylogicand
generalizedquantifiers.Alongtheway,theauthoralsoprovescompletenesstheorems
for many logics, including the cofinality quantifier logic of Shelah, a fully compact
extensionoffirst-orderlogic.
Withover500exercises,thisbookisidealforgraduatecourses,coveringthebasic
materialaswellasmoreadvancedapplications.
JoukoVäänänenisaProfessorofMathematicsattheUniversityofHelsinki, anda
ProfessorofMathematicalLogicandFoundationsofMathematicsattheUniversityof
Amsterdam.
CAMBRIDGESTUDIESINADVANCEDMATHEMATICS
EditorialBoard:
B.Bollobás,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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Models and Games
JOUKO VÄÄNÄNEN
UniversityofHelsinki
UniversityofAmsterdam
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521518123
©J.Väänänen2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-51812-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
ToJuliette

Contents
Preface pagexi
1 Introduction 1
2 PreliminariesandNotation 3
2.1 FiniteSequences 3
2.2 Equipollence 5
2.3 Countablesets 6
2.4 Ordinals 7
2.5 Cardinals 9
2.6 AxiomofChoice 10
2.7 HistoricalRemarksandReferences 11
Exercises 11
3 Games 14
3.1 Introduction 14
3.2 Two-PersonGamesofPerfectInformation 14
3.3 TheMathematicalConceptofGame 20
3.4 GamePositions 21
3.5 InfiniteGames 24
3.6 HistoricalRemarksandReferences 28
Exercises 28
4 Graphs 35
4.1 Introduction 35
4.2 First-OrderLanguageofGraphs 35
4.3 TheEhrenfeucht–Fra¨ısse´ GameonGraphs 38
4.4 Ehrenfeucht–Fra¨ısse´ GamesandElementaryEquivalence 43
4.5 HistoricalRemarksandReferences 48
Exercises 49
viii Contents
5 Models 53
5.1 Introduction 53
5.2 BasicConcepts 54
5.3 Substructures 62
5.4 Back-and-ForthSets 63
5.5 TheEhrenfeucht–Fra¨ısse´ Game 65
5.6 Back-and-ForthSequences 69
5.7 HistoricalRemarksandReferences 71
Exercises 71
6 First-OrderLogic 79
6.1 Introduction 79
6.2 BasicConcepts 79
6.3 CharacterizingElementaryEquivalence 81
6.4 TheLo¨wenheim–SkolemTheorem 85
6.5 TheSemanticGame 93
6.6 TheModelExistenceGame 98
6.7 Applications 102
6.8 Interpolation 107
6.9 UncountableVocabularies 113
6.10 Ultraproducts 119
6.11 HistoricalRemarksandReferences 125
Exercises 126
7 InfinitaryLogic 139
7.1 Introduction 139
7.2 PreliminaryExamples 139
7.3 TheDynamicEhrenfeucht–Fra¨ısse´ Game 144
7.4 SyntaxandSemanticsofInfinitaryLogic 157
7.5 HistoricalRemarksandReferences 170
Exercises 171
8 ModelTheoryofInfinitaryLogic 176
8.1 Introduction 176
8.2 Lo¨wenheim–SkolemTheoremforL 176
∞ω
8.3 ModelTheoryofL 179
ω1ω
8.4 LargeModels 184
8.5 ModelTheoryofL 191
κ+ω
8.6 GameLogic 201
8.7 HistoricalRemarksandReferences 222
Exercises 223