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
|
1.8 |
Additional Exercises |
23
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
Appendixes
|
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
|
B. Logic as the Algebra of Parts |
220
|
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
|
Bibliography |
253
|
Index |
257
|