Sets, logic and categories

Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

1 Naïve set theory	p. 1
1.1 Handle with care	p. 2
1.2 Basic deinitions	p. 4
1.3 Cartesian products, relations and functions	p. 7
1.4 IA Equivalence and order	p. 11
1.5 Bisections	p. 15
1.6 Finite sots	p. 20
1.7 Countable sots	p. 24
1.8 'The number systems	p. 28
1.9 Shoos and socks	p. 31
2 Ordinal numbers	p. 37
2.1 Well-order and induction	p. 38
2.2 The ordinals	p. 39
2.3 The hierarchy of sets	p. 47
2.4 Ordinal arithmetic	p. 49
3 Logic	p. 55
3.1 Formal logic	p. 56
3.2 Prepositional logic	p. 58
3.3 Soundness and completeness	p. 64
3.4 Boolean algebra	p. 69
4 First-order logic	p. 79
4.1 Language and syntax	p. 80
4.2 Semantics	p. 83
4.3 Deduction	p. 85
4.4 Soundness and completeness	p. 88
5 Model theory	p. 95
5.1 Compactness and Löwenheim-Skolcm	p. 95
5.2 Categoricily	p. 98
5.3 Peano arithmetic	p. 101
5.4 Consistency	p. 109
6 Axiomatic set theory	p. 113
6.1 Axioms for set thoorv	p. 114
6.2 The Axiom of Choice	p. 118
6.3 Cardinals	p. 124
0.4 Inaccessibility	p. 130
6.5 Alternative set theories	p. 133
6.6 The Skoiem Paradox	p. 134
6.7 Classes	p. 136
7 Categories	p. 141
7.1 Categories	p. 143
7.2 Foundations	p. 146
7.3 Functors	p. 148
7.4 Natural transformations	p. 150
8 Where to from here?	p. 155
8.1 Philosophy of mathematics	p. 155
8.2 Further reading	p. 58
Solutions to selected exercises	p. 161
References	p. 175
Index	p. 177

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