## Sets for MathematicsAdvanced 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 |

257 | |

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### Common terms and phrases

A₁ abstract sets Additional Exercises adjoint algebra arbitrary maps arrows axiom of choice Boolean called Cantor category of abstract category of sets characteristic function codomain cograph colimits commutative composition consider construction defined Definition denoted diagonal diagram domain dual elements endomap epimorphism equal equations equivalent example exists exponentiation fiber finite limits functor given graph hence Hint homomorphism idempotent implies injective inverse image isomorphism linear logic M-sets mapping ƒ mathematical Maximal Principle means monoid monomapping monomorphism morphisms natural transformation nilpotent notation one-element operation pair parameterization partially ordered sets poset Proof Proposition prove pullback real numbers relation retraction right A-set rule of inference satisfies set theory sets and mappings Show slice category specific statement structure surjective t₁ terminal object theorem topos true truth values unique map universal mapping property variable sets vector space X₁ Y₁