Sets for Mathematics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Why Sets for Mathematics?......Page 11
Organization......Page 12
Contributors to Sets for Mathematics......Page 15
1.1 Sets, Mappings, and Composition......Page 17
1.2 Listings, Properties, and Elements......Page 20
1.3 Surjective and Injective Mappings......Page 24
1.4 Associativity and Categories......Page 26
1.5 Separators and the Empty Set......Page 27
1.6 Generalized Elements......Page 31
1.7 Mappings as Properties......Page 33
1.8 Additional Exercises......Page 39
2.1 Sum as a Universal Property......Page 42
2.2 Monomorphisms and Parts......Page 48
2.3 Inclusion and Membership......Page 50
2.4 Characteristic Functions......Page 54
2.5 Inverse Image of a Part......Page 56
INVERSE IMAGE OF A PART ALONG A MAP......Page 57
2.6 Additional Exercises......Page 60
3.1 Retractions......Page 64
3.2 Isomorphism and Dedekind Finiteness......Page 70
FINITE INVERSE LIMITS......Page 74
3.4 Equalizers......Page 82
3.5 Pullbacks......Page 85
3.6 Inverse Limits......Page 87
3.7 Additional Exercises......Page 91
4.1 Colimits are Dual to Limits......Page 94
4.2 Epimorphisms and Split Surjections......Page 96
4.3 The Axiom of Choice......Page 100
4.4 Partitions and Equivalence Relations......Page 101
4.5 Split Images......Page 105
4.6 The Axiom of Choice as the Distinguishing Property of Constant/Random Sets......Page 108
4.7 Additional Exercises......Page 110
5.1 Natural Bijection and Functoriality......Page 112
5.2 Exponentiation......Page 114
Operators or Functionals......Page 117
5.3 Functoriality of Function Spaces......Page 118
5.4 Additional Exercises......Page 124
6.1 Axioms for Abstract Sets and Mappings......Page 127
6.2 Truth Values for Two-Stage Variable Sets......Page 130
6.3 Additional Exercises......Page 133
7.1 Concrete Duality: The Behavior of Monics and Epics under the Contravariant Functoriality of Exponentiation......Page 136
7.2 The Distributive Law......Page 142
7.3 Cantor’s Diagonal Argument......Page 145
7.4 Additional Exercises......Page 150
8.1 Images......Page 152
8.2 The Covariant Power Set Functor......Page 157
8.3 The Natural Map…......Page 161
8.4 Measuring, Averaging, and Winning with V-Valued Quantities......Page 164
8.5 Additional Exercises......Page 168
9.1 The Axiom of Infinity: Number Theory......Page 170
9.2 Recursion......Page 173
9.3 Arithmetic of N......Page 176
9.4 Additional Exercises......Page 181
10.1 Monoids, Posets, and Groupoids......Page 183
10.2.1 Actions as a Typical Model of Additional Variation and the Failure of the Axiom of Choice as Typical......Page 187
10.3 Reversible Graphs......Page 192
10.4 Chaotic Graphs......Page 196
10.5 Feedback and Control......Page 202
10.6 To and from Idempotents......Page 205
10.7 Additional Exercises......Page 207
A.0 Why Study Logic?......Page 209
A.1 Basic Operators and Their Rules of Inference......Page 211
A.2 Fields, Nilpotents, Idempotents Examples of Logical Operators in Algebra......Page 228
Appendix B The Axiom of Choice and Maximal Principles......Page 236
Adjoint Functors:......Page 247
Algebraic Topology:......Page 248
Composition:......Page 249
Converse of an Implication:......Page 250
Foundation, Category of Categories as:......Page 251
Functor (10.18):......Page 252
Inverse Image (2.5):......Page 253
Left-Cancellation / Monomapping (2.6):......Page 254
Logic, Objective:......Page 255
Mapping (1.1):......Page 256
One-Element Set (Section 1.2):......Page 257
Set Theory:......Page 258
Set Theory, Parameterization:......Page 259
Topos:......Page 260
Topos and the Cantorian Contrast:......Page 261
Topos of Abstract Sets:......Page 262
Union:......Page 263
Yoneda Embedding:......Page 264
Yoneda’s Lemma:......Page 265
Yoneda and Totality:......Page 266
C.2 Mathematical Notations and Logical Symbols......Page 267
C.3 The Greek Alphabet......Page 268
Bibliography......Page 269
Additional References......Page 271
Index......Page 273