# Sets for Mathematics (Book)

Information Basic Mathematics F. William Lawvere English Cambridge University Press 10 April 2003 276 0521010608 978-0521010603

The textbook Sets for Mathematics by F. William Lawvere uses categorical algebra to introduce set theory.

Chapter/Section # Title Page #
Foreword ix
Contributors to Sets for Mathematics xiii
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
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
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
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
5. Mapping Sets and Exponentials
5.1 Natural Bijection and Functoriality 96
5.2 Exponentiation 98
5.3 Functoriality of Function Spaces 102
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
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
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
9. Introduction to Variable Sets
9.1 The Axiom of Infinity: Number Theory 154
9.2 Recursion 157
9.3 Arithmetic of \(N\) 160
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