Lectures in Logic and Set Theory. Volume 1: Mathematical Logic

342 Pages·2003·1.38 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
LecturesinLogicandSetTheoryVolume1
Thistwo-volumeworkbridgesthegapbetweenintroductoryexpositionsof
logicorsettheoryononehand,andtheresearchliteratureontheother.Itcan
beusedasatextinanadvancedundergraduateorbeginninggraduatecourse
inmathematics,computerscience,orphilosophy.Thevolumesarewrittenin
auser-friendlyconversationallecturestylethatmakesthemequallyeffective
forself-studyorclassuse.
Volume1includesformalprooftechniques,asectiononapplicationsof
compactness(includingnon-standardanalysis),agenerousdoseofcomputa-
bilityanditsrelationtotheincompletenessphenomenon,andthefirstpresen-
tationofacompleteproofofGo¨del’ssecondincompletenesstheoremsince
HilbertandBernay’sGrundlagen.
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
Volume1:MathematicalLogic
GEORGE TOURLAKIS
YorkUniversity
  
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Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States by Cambridge University Press, New York
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© 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-13 978-0-511-06871-3 eBook (EBL)
ISBN-10 0-511-06871-9 eBook (EBL)
ISBN-13 978-0-521-75373-9 hardback
ISBN-10 0-521-75373-2 hardback
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Contents
Preface pageix
I BasicLogic 1
I.1 FirstOrderLanguages 5
I.2 ADigressionintotheMetatheory:
InformalInductionandRecursion 19
I.3 AxiomsandRulesofInference 28
I.4 BasicMetatheorems 42
I.5 Semantics;Soundness,Completeness,Compactness 52
I.6 Substructures,Diagrams,andApplications 75
I.7 DefinedSymbols 112
I.8 ComputabilityandUncomputability 123
I.9 Arithmetic,Definability,Undefinability,
andIncompletableness 155
I.10 Exercises 191
II TheSecondIncompletenessTheorem 205
II.1 PeanoArithmetic 206
II.2 AFormalβ-Function 232
II.3 FormalPrimitiveRecursion 248
II.4 TheBoldface(cid:14)and(cid:15) 256
II.5 Arithmetization 265
II.6 DerivabilityConditions;FixedPoints 272
II.7 Exercises 316
Bibliography 319
ListofSymbols 321
Index 323
vii