Lectures in Logic and Set Theory. Volume 2: Set Theory

593 Pages·2003·2.13 MB·English

by  George Tourlakis

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CAMBRIDGE STUDIES IN
ADVANCED MATHEMATICS
EDITORIAL BOARD
B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK
LecturesinLogicandSetTheoryVolume2
Thistwo-volumeworkbridgesthegapbetweenintroductoryexpositionsof
logicorsettheoryononehand,andtheresearchliteratureontheother.Itcan
beusedasatextinanadvancedundergraduateorbeginninggraduatecourse
inmathematics,computerscience,orphilosophy.Thevolumesarewrittenin
auser-friendlyconversationallecturestylethatmakesthemequallyeffective
forself-studyorclassuse.
Volume 2, on formal (ZFC) set theory, incorporates a self-contained
“Chapter0”onprooftechniquessothatitisbasedonformallogic,inthestyle
ofBourbaki.Theemphasisonbasictechniqueswillprovidethereaderwith
asolidfoundationinsettheoryandsetsacontextforthepresentationofad-
vancedtopicssuchasabsoluteness,relativeconsistencyresults,twoexposi-
tionsofGo¨del’sconstructibleuniverse,numerouswaysofviewingrecursion,
andachapteronCohenforcing.
George Tourlakis is Professor of Computer Science at York University of
Ontario.
Alreadypublished
2 K.PetersenErgodictheory
3 P.T.JohnstoneStonespaces
5 J.-P.KahaneSomerandomseriesoffunctions,2ndedition
7 J.Lambek&P.J.ScottIntroductiontohigher-ordercategoricallogic
8 H.MatsumuraCommutativeringtheory
10 M.AschbacherFinitegrouptheory,2ndedition
11 J.L.AlperinLocalrepresentationtheory
12 P.KoosisThelogarithmicintegralI
14 S.J.PattersonAnintroductiontothetheoryoftheRiemannzeta-function
15 H.J.BauesAlgebraichomotopy
16 V.S.VaradarajanIntroductiontoharmonicanalysisonsemisimpleLiegroups
17 W.Dicks&M.DunwoodyGroupsactingongraphs
19 R.Fritsch&R.PiccininiCellularstructuresintopology
20 H.KlingenIntroductorylecturesonSiegelmodularforms
21 P.KoosisThelogarithmicintegralII
22 M.J.CollinsRepresentationsandcharactersoffinitegroups
24 H.KunitaStochasticflowsandstochasticdifferentialequations
25 P.WojtaszczykBanachspacesforanalysts
26 J.E.Gilbert&M.A.M.MurrayCliffordalgebrasandDiracoperatorsinharmonicanalysis
27 A.Frohlich&M.J.TaylorAlgebraicnumbertheory
28 K.Goebel&W.A.KirkTopicsinmetricfixedpointtheory
29 J.F.HumphreysReflectiongroupsandCoxetergroups
30 D.J.BensonRepresentationsandcohomologyI
31 D.J.BensonRepresentationsandcohomologyII
32 C.Allday&V.PuppeCohomologicalmethodsintransformationgroups
33 C.Souleetal.LecturesonArakelovgeometry
34 A.Ambrosetti&G.ProdiAprimerofnonlinearanalysis
35 J.Palis&F.TakensHyperbolicity,stabilityandchaosathomoclinicbifurcations
37 Y.MeyerWaveletsandoperators1
38 C.Weibel,Anintroductiontohomologicalalgebra
39 W.Bruns&J.HerzogCohen-Macaulayrings
40 V.SnaithExplicitBrauerinduction
41 G.LaumonCohomologyofDrinfeldmodularvarietiesI
42 E.B.DaviesSpectraltheoryanddifferentialoperators
43 J.Diestel,H.Jarchow,&A.TongeAbsolutelysummingoperators
44 P.MattilaGeometryofsetsandmeasuresinEuclideanspaces
45 R.PinskyPositiveharmonicfunctionsanddiffusion
46 G.TenenbaumIntroductiontoanalyticandprobabilisticnumbertheory
47 C.PeskineAnalgebraicintroductiontocomplexprojectivegeometry
48 Y.Meyer&R.CoifmanWavelets
49 R.StanleyEnumerativecombinatoricsI
50 I.PorteousCliffordalgebrasandtheclassicalgroups
51 M.AudinSpinningtops
52 V.JurdjevicGeometriccontroltheory
53 H.VolkleinGroupsasGaloisgroups
54 J.LePotierLecturesonvectorbundles
55 D.BumpAutomorphicformsandrepresentations
56 G.LaumonCohomologyofDrinfeldmodularvarietiesII
57 D.M.Clark&B.A.DaveyNaturaldualitiesfortheworkingalgebraist
58 J.McClearyAuser’sguidetospectralsequencesII
59 P.TaylorPracticalfoundationsofmathematics
60 M.P.Brodmann&R.Y.SharpLocalcohomology
61 J.D.Dixonetal.Analyticpro-Pgroups
62 R.StanleyEnumerativecombinatoricsII
63 R.M.DudleyUniformcentrallimittheorems
64 J.Jost&X.Li-JostCalculusofvariations
65 A.J.Berrick&M.E.KeatingAnintroductiontoringsandmodules
66 S.MorosawaHolomorphicdynamics
67 A.J.Berrick&M.E.KeatingCategoriesandmoduleswithK-theoryinview
68 K.SatoLevyprocessesandinfinitelydivisibledistributions
69 H.HidaModularformsandGaloiscohomology
70 R.Iorio&V.IorioFourieranalysisandpartialdifferentialequations
71 R.BleiAnalysisinintegerandfractionaldimensions
72 F.Borceaux&G.JanelidzeGaloistheories
73 B.BollobasRandomgraphs
LECTURES IN LOGIC
AND SET THEORY
Volume2:SetTheory
GEORGE TOURLAKIS
YorkUniversity
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521753746
© George Tourlakis 2003
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 2003
ISBN-13 978-0-511-06872-0 eBook (EBL)
ISBN-10 0-511-06872-7 eBook (EBL)
ISBN-13 978-0-521-75374-6 hardback
ISBN-10 0-521-75374-0 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
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Tothememoryofmyparents

Contents
Preface pagexi
I ABitofLogic:AUser’sToolbox 1
I.1 FirstOrderLanguages 7
I.2 ADigressionintotheMetatheory:
InformalInductionandRecursion 20
I.3 AxiomsandRulesofInference 29
I.4 BasicMetatheorems 43
I.5 Semantics 53
I.6 DefinedSymbols 66
I.7 FormalizingInterpretations 77
I.8 TheIncompletenessTheorems 87
I.9 Exercises 94
II TheSet-TheoreticUniverse,Na¨ıvely 99
II.1 The“RealSets” 99
II.2 ANa¨ıveLookatRussell’sParadox 105
II.3 TheLanguageofAxiomaticSetTheory 106
II.4 OnNames 110
III TheAxiomsofSetTheory 114
III.1 Extensionality 114
III.2 SetTerms;Comprehension;Separation 119
III.3 TheSetofAllUrelements;theEmptySet 130
III.4 ClassTermsandClasses 134
III.5 AxiomofPairing 145
III.6 AxiomofUnion 149
III.7 AxiomofFoundation 156
III.8 AxiomofCollection 160
III.9 AxiomofPowerSet 178
vii
viii Contents
III.10 PairingFunctionsandProducts 182
III.11 RelationsandFunctions 193
III.12 Exercises 210
IV TheAxiomofChoice 215
IV.1 Introduction 215
IV.2 MoreJustificationforAC;the“Constructible”
UniverseViewpoint 218
IV.3 Exercises 229
V TheNaturalNumbers;TransitiveClosure 232
V.1 TheNaturalNumbers 232
V.2 AlgebraofRelations;TransitiveClosure 253
V.3 AlgebraofFunctions 272
V.4 EquivalenceRelations 276
V.5 Exercises 281
VI Order 284
VI.1 POClasses,LOClasses,andWOClasses 284
VI.2 InductionandInductiveDefinitions 293
VI.3 ComparingOrders 316
VI.4 Ordinals 323
VI.5 TheTransfiniteSequenceofOrdinals 340
VI.6 ThevonNeumannUniverse 358
VI.7 APairingFunctionontheOrdinals 373
VI.8 Absoluteness 377
VI.9 TheConstructibleUniverse 395
VI.10 ArithmeticontheOrdinals 410
VI.11 Exercises 426
VII Cardinality 430
VII.1 Finitevs.Infinite 431
VII.2 EnumerableSets 442
VII.3 Diagonalization;UncountableSets 451
VII.4 Cardinals 457
VII.5 ArithmeticonCardinals 470
VII.6 Cofinality;MoreCardinalArithmetic;
InaccessibleCardinals 478
VII.7 InductivelyDefinedSetsRevisited;
RelativeConsistencyofGCH 494
VII.8 Exercises 512
VIII Forcing 518
VIII.1 POSets,Filters,andGenericSets 520
VIII.2 ConstructingGenericExtensions 524