# Book:Alan G. Hamilton/Numbers, Sets and Axioms

## Alan G. Hamilton: Numbers, Sets and Axioms: The Apparatus of Mathematics

Published $1982$, Cambridge University Press

ISBN 0-521-28761-8.

### Contents

Preface
1 Numbers
1.1 Natural numbers and integers
1.2 Rational numbers
1.3 Real numbers
1.4 Decimal notation
2 The size of a set
2.1 Finite and countable sets
2.2 Uncountable sets
2.3 Cardinal numbers
3 Ordered sets
3.1 Order relations and ordered sets
3.2 Lattices and Boolean algebras
4 Set theory
4.1 What is a set?
4.2 The Zermelo-Fraenkel axioms
4.3 Mathematics in ZF
4.4 Sets and classes
4.5 Models of set theory
5 The axiom of choice
5.1 The axiom of choice and direct applications
5.2 Zorn's lemma and the well-ordering theorem
5.3 Other consequences of the axiom of choice
6 Ordinal and cardinal numbers
6.1 Well-ordered sets and ordinal numbers
6.2 Transfinite recursion and ordinal arithmetic
6.3 Cardinal numbers
Hints and solutions to selected exercises
References
Index of symbols
Subject index

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