Sets for Mathematics (Book): Difference between revisions

|image=[[File:Lawvere Sets for Mathematics Cover.jpg]]

|author=[https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere]

|publisher=Cambridge University Press

|publicationdate=10 April 2003

|pages=276

|isbn10=0521010608

|isbn13=978-0521010603

The textbook '''''Sets for Mathematics''''' by [https://en.wikipedia.org/wiki/William_Lawvere F. William Lawvere] uses categorical algebra to introduce set theory.

! colspan="2" | Foreword || ix

! colspan="2" | Contributors to Sets for Mathematics || xiii

! colspan="3" | 1. Abstract Sets and Mappings

| 1.1 || Sets, Mappings, and Composition || 1

| 1.2 || Listings, Properties, and Elements || 4

| 1.3 || Surjective and Injective Mappings || 8

| 1.4 || Associativity and Categories || 10

| 1.5 || Separators and the Empty Set || 11

| 1.6 || Generalized Elements || 15

| 1.7 || Mappings as Properties || 17

|-  

| 1.8 || Additional Exercises || 23

|-  

! colspan="3" | 2. Sums, Monomorphisms, and Parts

| 2.1 || Sum as a Universal Property || 26

| 2.2 || Monomorphisms and Parts || 32

| 2.3 || Inclusion and Membership || 34

| 2.4 || Characteristic Functions || 38

| 2.5 || Inverse Image of a Part || 40

| 2.6 || Additional Exercises || 44

! colspan="3" | 3. Finite Inverse Limits

| 3.1 || Retractions || 48

| 3.2 || Isomorphism and Dedekind Finiteness || 54

| 3.3 || Cartesian Products and Graphs || 58

| 3.4 || Equalizers || 66

| 3.5 || Pullbacks || 69

| 3.6 || Inverse Limits || 71

| 3.7 || Additional Exercises || 75

! colspan="3" | Colimits, Epimorphisms, and the Axiom of Choice

| 4.1 || Colimits are Dual to Limits || 78

| 4.2 || Epimorphisms and Split Surjections || 80

| 4.3 || The Axiom of Choice || 84

| 4.4 || Partitions and Equivalence Relations || 85

| 4.5 || Split Images || 89

| 4.6 || The Axiom of Choice as the Distinguishing Property of Constant/Random Sets || 92

| 4.7 || Additional Exercises || 94

! colspan="3" | 5. Mapping Sets and Exponentials

| 5.1 || Natural Bijection and Functoriality || 96

| 5.2 || Exponentiation || 98

| 5.3 || Functoriality of Function Spaces || 102

| 5.4 || Additional Exercises || 108

! colspan="3" | 6. Summary of the Axioms and an Example of Variable Sets

| 6.1 || Axioms for Abstract Sets and Mappings || 111

| 6.2 || Truth Values for Two-Stage Variable Sets || 114

| 6.3 || Additional Exercises || 117

! colspan="3" | 7. Consequences and Uses of Exponentials

| 7.1 || Concrete Duality: The Behavior of Monics and Epics under the Contravariant Functoriality of Exponentiation || 120

| 7.2 || The Distributive Law || 126

| 7.3 || Cantor's Diagonal Argument || 129

| 7.4 || Additional Exercises || 134

! colspan="3" | 8. More on Power Sets

| 8.1 || Images || 136

| 8.2 || The Covariant Power Set Functor || 141

| 8.3 || The Natural Map \(Placeholder\) || 145

| 8.4 || Measuring, Averaging, and Winning with \(V\)-Valued Quantities || 148

| 8.5 || Additional Exercises || 152

! colspan="3" | 9. Introduction to Variable Sets

| 9.1 || The Axiom of Infinity: Number Theory || 154

| 9.2 || Recursion || 157

| 9.3 || Arithmetic of \(N\) || 160

| 9.4 || Additional Exercises || 165

! colspan="3" | 10. Models of Additional Variation

| 10.1 || Monoids, Podsets, and Groupoids || 167

| 10.2 || Actions || 171

| 10.3 || Reversible Graphs || 176

| 10.4 || Chaotic Graphs || 180

| 10.5 || Feedback and Control || 186

| 10.6 || To and from Idempotents || 189

| 10.7 || Additional Exercises || 191

! colspan="3" | Appendixes

! colspan="3" | A. Logic as the Algebra of Parts

| A.0 || Why Study Logic? || 193

| A.1 || Basic Operators and Their Rules of Inference || 195

| A.2 || Fields, Nilpotents, Idempotents || 212

! colspan="2" | B. Logic as the Algebra of Parts || 220

! colspan="3" | C. Definitions, Symbols, and the Greek Alphabet

| C.1 || Definitions of Some Mathematical and Logical Concepts || 231

| C.2 || Mathematical Notations and Logical Symbols || 251

| C.3 || The Greek Alphabet || 252

! colspan="2" | Bibliography || 253

! colspan="2" | Index || 257