Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism

365 Pages·2018·2.194 MB·English

by  John T. Baldwin

ModelTheoryandthePhilosophyof
MathematicalPractice
Major shifts in the field of model theory in the twentieth century
have seen the development of new tools, methods, and motivations
formathematicians and philosophers. In this book, John T. Baldwin
placestherevolutioninitshistoricalcontextfromtheancientGreeks
to the last century, argues for local rather than global foundations
for mathematics, and provides philosophical viewpoints on the
importance of modern model theory for both understanding and
undertaking mathematical practice. The volume also addresses the
impactofmodeltheoryoncontemporaryalgebraicgeometry,number
theory,combinatorics,anddifferentialequations.Thiscomprehensive
and detailed book will interest logicians and mathematicians as well
asthoseworkingonthehistoryandphilosophyofmathematics.
johnt.baldwinisProfessorEmeritusintheDepartmentofMath-
ematics,Statistics,andComputerScienceattheUniversityofIllinois
atChicago.Hehaspublishedwidelyonmathematicsandphilosophy,
andheistheauthorofanumberofbooksincludingFundamentalsof
StabilityTheory(1988)andCategoricity(2009).

Model Theory and the Philosophy
of Mathematical Practice
Formalization without Foundationalism
john t baldwin
.
UniversityofIllinois,Chicago
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Informationonthistitle:www.cambridge.org/9781107189218
DOI:10.1017/9781316987216
©JohnT.Baldwin2018
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To:
Grace,Sharon,Katie,PoppyGrace
Mother,wife,daughter,granddaughter

Contents
ListofFigures [pagex]
Acknowledgments [xi]
Introduction [1]
part i refining the notion of categoricity [29]
1 Formalization [31]
1.1 TheConceptofFormalization [32]
1.2 VocabularyandStructures [34]
1.3 Logics [41]
1.4 TheoriesandAxioms [47]
2 TheContextofFormalization [51]
2.1 TheProcessofFormalization [51]
2.2 TwoRolesofFormalization [54]
2.3 ACriterionforEvaluatingPropertiesofTheories [58]
2.4 VirtuousPropertiesasanOrganizingPrinciple [61]
3 Categoricity [68]
3.1 CategoricityofSecondOrderTheories [69]
3.2 Lω1,ω-categoricity [74]
3.3 Lω,ω:CategoricityinPower [78]
3.4 TheSignificanceofCategoricity(inPower) [84]
part ii the paradigm shift [87]
4 WhatWasModelTheoryAbout? [89]
4.1 TheDownwardLo¨wenheim–Skolem–TarskiTheorem [89]
4.2 Completeness,Compactness,andtheUpward
Lo¨wenheim–Skolem–TarskiTheorem [92]
4.3 CompleteTheories [99]
4.4 QuantifierComplexity [104]
4.5 Interpretability [108]
4.6 WhatIsaStructure,Really? [111]
4.7 WhenAreStructures‘Equal’? [116]
vii
viii contents
5 WhatIsContemporaryModelTheoryAbout? [119]
5.1 AnalogytoTheoremtoMethod [119]
5.2 UniversalDomains [124]
5.3 TheStabilityHierarchy [128]
5.4 CombinatorialGeometry [133]
5.5 Classification:TheMainGap [137]
5.6 WhyIsModelTheorySoEntwinedwith
ClassicalMathematics? [143]
6 IsolatingTameMathematics [148]
6.1 GroupsofFiniteMorleyRank [149]
6.2 FormalMethodsasaToolinMathematics [151]
6.3 FirstOrderAnalysis [156]
6.4 WhatAretheCentralNotionsofModelTheory? [162]
7 InfinitaryLogic [167]
7.1 CategoricityinUncountablePowerforLω1,ω [168]
7.2 TheVaughtConjecture [171]
7.3 De´javu:CategoricityinInfinitarySecondOrderLogic [175]
8 ModelTheoryandSetTheory [177]
8.1 IsThereModelTheorywithoutAxiomaticSetTheory? [178]
8.2 IsThereModelTheorywithoutCombinatorialSetTheory? [182]
8.3 WhyIsℵ0ExceptionalforModelTheory? [186]
8.4 EntanglementofModelTheoryandCardinality [189]
8.5 EntanglementofModelTheoryandtheReplacementAxiom [192]
8.6 EntanglementofModelTheorywithExtensionsofZFC [196]
8.7 Moral [198]
part iii geometry [201]
9 AxiomatizationofGeometry [203]
9.1 TheGoalsofAxiomatization [205]
9.2 DescriptionsoftheGeometricContinuum [211]
9.3 SomeGeometricDataSetsandAxiomSystems [217]
9.4 GeometryandAlgebra [221]
9.5 ProportionandArea [229]
10 π,Area,andCircumferenceofCircles [234]
10.1 πinEuclideanandArchimedeanGeometry [234]
10.2 FromDescartestoTarski [239]
10.3 πinGeometriesoverRealClosedFields [243]
11 Complete:TheWordforAllSeasons [250]
11.1 Hilbert’sContinuityAxioms [252]
11.2 AgainsttheDedekindPostulateforGeometry [255]
contents ix
part iv methodology [259]
12 FormalizationandPurityinGeometry [261]
12.1 ContentandVocabulary [262]
12.2 ProjectiveandAffineGeometry [265]
12.3 GeneralSchemesforCharacterizingPurity [267]
12.4 Modesty,Purity,andGeneralization [273]
12.5 PurityandtheDesarguesProposition [273]
12.6 DistinguishingAlgebraicandGeometricProof [281]
13 OntheNatureofDefinition:ModelTheory [283]
13.1 MethodologyofClassification [285]
13.2 TheFecundityoftheStabilityHierarchy [287]
13.3 DividingLines [292]
13.4 Definition,Classification,andTaxonomy [294]
14 Formalism-Freeness(MathematicalProperties) [300]
15 Summation [312]
References [317]
Index [347]