Sets for mathematics

276 Pages·2003·1.772 MB·English

by  F. William Lawvere, Robert Rosebrugh

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SETS FOR MATHEMATICS
Advancedundergraduateorbeginninggraduatestudentsneedaunifiedfoundation
fortheirstudyofmathematics.Forthefirsttimeinatext,thisbookusescategorical
algebratobuildsuchafoundation,startingfromintuitivedescriptionsofmathemat-
icallyandphysicallycommonphenomenaandadvancingtoaprecisespecification
ofthenatureofcategoriesofsets.
Settheoryasthealgebraofmappingsisintroducedanddevelopedasaunifying
basisforadvancedmathematicalsubjectssuchasalgebra,geometry,analysis,and
combinatorics.Theformalstudyevolvesfromgeneralaxiomsthatexpressuniver-
salpropertiesofsums,products,mappingsets,andnaturalnumberrecursion.The
distinctive features of Cantorian abstract sets, as contrasted with the variable and
cohesivesetsofgeometryandanalysis,aremadeexplicitandtakenasspecialax-
ioms.Functorcategoriesareintroducedtomodelthevariablesetsusedingeometry
andtoillustratethefailureoftheaxiomofchoice.Anappendixprovidesanexplicit
introductiontonecessaryconceptsfromlogic,andanextensiveglossaryprovides
awindowtothemathematicallandscape.
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SETS FOR MATHEMATICS
F. WILLIAM LAWVERE
StateUniversityofNewYorkatBuffalo
ROBERT ROSEBRUGH
MountAllisonUniversity
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  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© F. William Lawvere, Robert Rosebrugh 2003
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First published in print format 2003
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Contents
Foreword page ix
ContributorstoSetsforMathematics xiii
1 AbstractSetsandMappings 1
1.1 Sets, Mappings, and Composition 1
1.2 Listings,Properties,andElements 4
1.3 SurjectiveandInjectiveMappings 8
1.4 Associativity and Categories 10
1.5 SeparatorsandtheEmptySet 11
1.6 GeneralizedElements 15
1.7 MappingsasProperties 17
1.8 AdditionalExercises 23
2 Sums,Monomorphisms,andParts 26
2.1 SumasaUniversalProperty 26
2.2 MonomorphismsandParts 32
2.3 InclusionandMembership 34
2.4 CharacteristicFunctions 38
2.5 InverseImageofaPart 40
2.6 AdditionalExercises 44
3 FiniteInverseLimits 48
3.1 Retractions 48
3.2 IsomorphismandDedekindFiniteness 54
3.3 CartesianProductsandGraphs 58
3.4 Equalizers 66
3.5 Pullbacks 69
3.6 InverseLimits 71
3.7 AdditionalExercises 75
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4 Colimits,Epimorphisms,andtheAxiomofChoice 78
4.1 ColimitsareDualtoLimits 78
4.2 EpimorphismsandSplitSurjections 80
4.3 TheAxiomofChoice 84
4.4 PartitionsandEquivalenceRelations 85
4.5 SplitImages 89
4.6 TheAxiomofChoiceastheDistinguishing Property
ofConstant/RandomSets 92
4.7 AdditionalExercises 94
5 MappingSetsandExponentials 96
5.1 NaturalBijectionandFunctoriality 96
5.2 Exponentiation 98
5.3 Functoriality of Function Spaces 102
5.4 AdditionalExercises 108
6 Summary of the Axioms and an Example of Variable Sets 111
6.1 AxiomsforAbstractSetsandMappings 111
6.2 TruthValuesforTwo-StageVariableSets 114
6.3 Additional Exercises 117
7 ConsequencesandUsesofExponentials 120
7.1 ConcreteDuality:TheBehaviorofMonicsandEpicsunder
theContravariantFunctorialityofExponentiation 120
7.2 TheDistributiveLaw 126
7.3 Cantor’s Diagonal Argument 129
7.4 Additional Exercises 134
8 MoreonPowerSets 136
8.1 Images 136
8.2 TheCovariantPowerSetFunctor 141
8.3 TheNaturalMapPX (cid:2)(cid:2)22X 145
8.4 Measuring,Averaging,andWinningwithV-ValuedQuantities 148
8.5 AdditionalExercises 152
9 IntroductiontoVariableSets 154
9.1 TheAxiomofInfinity:NumberTheory 154
9.2 Recursion 157
9.3 Arithmetic of N 160
9.4 AdditionalExercises 165
10 ModelsofAdditionalVariation 167
10.1 Monoids,Posets,andGroupoids 167
10.2 Actions 171
10.3 ReversibleGraphs 176
10.4 ChaoticGraphs 180
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Contents vii
10.5 FeedbackandControl 186
10.6 ToandfromIdempotents 189
10.7 AdditionalExercises 191
Appendixes 193
A LogicastheAlgebraofParts 193
A.0 WhyStudyLogic? 193
A.1 BasicOperatorsandTheirRulesofInference 195
A.2 Fields,Nilpotents,Idempotents 212
B TheAxiomofChoiceandMaximalPrinciples 220
C Definitions,Symbols,andtheGreekAlphabet 231
C.1 DefinitionsofSomeMathematicalandLogicalConcepts 231
C.2 MathematicalNotationsandLogicalSymbols 251
C.3 TheGreekAlphabet 252
Bibliography 253
Index 257
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Foreword
WhySetsforMathematics?
This book is for students who are beginning the study of advanced mathematical
subjectssuchasalgebra,geometry,analysis,orcombinatorics.Ausefulfoundation
forthesesubjectswillbeachievedbyopenlybringingoutandstudyingwhatthey
haveincommon.
A significant part of what is common to all these subjects was made explicit
100yearsagobyRichardDedekindandGeorgCantor,andanothersignificantpart
50 years ago by Samuel Eilenberg and Saunders Mac Lane. The resulting idea of
categoriesofsetsisthemaincontentofthisbook.Itisworththeefforttostudythis
ideabecauseitprovidesaunifiedguidetoapproachingconstructionsandproblems
inthescienceofspaceandquantity.
Morespecifically,ithasbecomestandardpracticetorepresentanobjectofmath-
ematicalinterest(forexampleasurfaceinthree-dimensionalspace)asa“structure.”
Thisrepresentationispossiblebymeansofthefollowingtwosteps:
(1) Firstwedepletetheobjectofnearlyallcontent.Wecouldthinkofanidealized
computer memory bank that has been erased, leaving only the pure locations
(thatcouldbefilledwithanynewdatathatarerelevant).Thebagofpurepoints
resulting from this process was called by Cantor a Kardinalzahl, but we will
usuallyrefertoitasanabstractset.
(2) Then,justascomputerscanbewiredupinspecificways,suitablespecificmap-
pings between these structureless sets will constitute a structure that reflects
the complicated content of a mathematical object. For example, the midpoint
operationinEuclideangeometryisrepresentedasamappingwhose“value”at
anypairofpointsisaspecialthirdpoint.
Toexplainthebasisforthesestepsthereisanimportantprocedureknownasthe
axiomaticmethod:Thatis,fromtheongoinginvestigationoftheideasofsetsand
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