# Sets for Mathematics

Cambridge University Press, Jan 27, 2003 - Mathematics - 261 pages
Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion.

### Contents

 Abstract Sets and Mappings 1 12 Listings Properties and Elements 4 13 Surjective and Injective Mappings 8 14 Associativity and Categories 10 15 Separators and the Empty Set 11 16 Generalized Elements 15 17 Mappings as Properties 17 18 Additional Exercises 23
 62 Truth Values for TwoStage Variable Sets 114 63 Additional Exercises 117 Consequences and Uses of Exponentials 120 72 The Distributive Law 126 73 Cantors Diagonal Argument 129 74 Additional Exercises 134 More on Power Sets 136 82 The Covariant Power Set Functor 141

 Sums Monomorphisms and Parts 26 22 Monomorphisms and Parts 32 23 Inclusion and Membership 34 24 Characteristic Functions 38 25 Inverse Image of a Part 40 26 Additional Exercises 44 Finite Inverse Limits 48 32 Isomorphism and Dedekind Finiteness 54 33 Cartesian Products and Graphs 58 34 Equalizers 66 35 Fullbacks 69 36 Inverse Limits 71 37 Additional Exercises 75 Colimits Epimorphisms and the Axiom of Choice 78 42 Epimorphisms and Split Surjections 80 43 The Axiom of Choice 84 44 Partitions and Equivalence Relations 85 45 Split Images 89 46 The Axiom of Choice as the Distinguishing Property of ConstantRandom Sets 92 47 Additional Exercises 94 Mapping Sets and Exponentials 96 52 Exponentiation 98 53 Functoriality of Function Spaces 102 54 Additional Exercises 108 Summary of the Axioms and an Example of Variable Sets 111
 83 The Natural Map PX2²ˣ 145 84 Measuring Averaging and Winning with VValued Quantities 148 85 Additional Exercises 152 Introduction to Variable Sets 154 92 Recursion 157 93 Arithmetic of N 160 94 Additional Exercises 165 Models of Additional Variation 167 102 Actions 171 103 Reversible Graphs 176 104 Chaotic Graphs 180 105 Feedback and Control 186 106 To and from Idempotents 189 107 Additional Exercises 191 Logic as the Algebra of Parts 193 A1 Basic Operators and Their Rules of Inference 195 A2 Fields Nilpotents Idempotents 212 The Axiom of Choice and Maximal Principles 220 Definitions Symbols and the Greek Alphabet 231 C2 Mathematical Notations and Logical Symbols 251 C3 The Greek Alphabet 252 Bibliography 253 Index 257 Copyright