Sets for Mathematics (Book)

Basic Mathematics
Information
Author	F. William Lawvere
Language	English
Publisher	Cambridge University Press
Publication Date	10 April 2003
Pages	276
ISBN-10	0521010608
ISBN-13	978-0521010603

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

In parallel to Grothendieck, Lawvere developed the notion of a topos as a collection of objects/points behaving as sets and arrows as maps between sets. The utility of this is for a characterization of sets via mappings only - there is a unique (equivalence class of) set(s) with one element that can alternatively be described as only receiving one map from each other set. The number of maps in the other direction count the elements of sets conversely. Two element sets are an instance of "subobject classifiers" in the topos of sets such that the maps into them correspond to subsets of the source set of the map. The language of toposes is particularly accessible here, and plays a universal role in modern mathematics, e.g. for toposes emerging from sets parametrized by a topological space (presheaf toposes) even inspiring functional programming languages such as Haskell due to the logical properties of toposes.

Table of Contents

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 [math]\displaystyle{ Placeholder }[/math]	145
8.4	Measuring, Averaging, and Winning with [math]\displaystyle{ V }[/math]-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 [math]\displaystyle{ N }[/math]	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