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

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## Alan G. Hamilton:

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

Published $1982$, **Cambridge University Press**

- ISBN 0-521-28761-8.

### Subject Matter

### 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*