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

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

Published $\text {1982}$, Cambridge University Press

ISBN 0-521-28761-8

Subject Matter

• Abstract Algebra

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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Source work progress

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