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Tables of Contents for Numbers, Sets and Axioms

Chapter/Section Title

Page #

Page Count

Preface

vii

Numbers

1

50

Natural numbers and integers

1

17

Rational numbers

18

8

Real numbers

26

17

Decimal notation

43

8

The size of a set

51

31

Finite and countable sets

51

12

Uncountable sets

63

10

Cardinal numbers

73

9

Ordered sets

82

26

Order relations and ordered sets

82

15

Lattices and Boolean algebras

97

11

Set theory

108

55

What is a set?

108

7

The Zermelo-Fraenkel axioms

115

14

Mathematics in ZF

129

16

Sets and classes

145

11

Models of set theory

156

7

The axiom of choice

163

29

The axiom of choice and direct applications

163

8

Zorn's lemma and the well-ordering theorem

171

13

Other consequences of the axiom of choice

184

8

Ordinal and cardinal numbers

192

45

Well-ordered sets and ordinal numbers

192

13

Transfinite recursion and ordinal arithmetic

205

16

Cardinal numbers

221

16

Hints and solutions to selected exercises

237

10

References

247

2

Index of symbols

249

3

Subject index

252