Classic Set Theory: For Guided Independent Study: Goldrei; D.C.: 9780412606106: Book: Technology: eCampus.com

Preface

vii

1 Introduction

(6)

1.1. Outline of the book

(2)

1.2 Assumed Knowledge

(4)

2 The Real Numbers

(26)

2.1 Introduction

(1)

2.2 DedeKind's construction

(9)

2.3 Alternative constructions

(11)

2.4 The rational numbers

(5)

3 The Natural Numbers

(33)

3.1 Introduction

(5)

3.2 The construction of the natural numbers

(10)

3.3 Arithmetic

(10)

3.4 Finite sets

(8)

4 The Zermelo-Fraenkel Axioms

(38)

4.1 Introduction

(5)

4.2 A formal language

(4)

4.3 Axioms 1 to 3

(6)

4.4 Axioms 4 to 6

(11)

4.5 Axioms 7 to 9

(12)

5 The Axiom of Choice

104

(23)

5.1 Introduction

104

(3)

5.2 The axiom of choice

107

(5)

5.3 The axiom of choice and mathematics

112

(5)

Zorn's lemma

117

(10)

6 Cardinals (without the Axiom of Choice)

127

(36)

6.1 Introduction

127

(1)

6.2 Comparing sizes

128

(5)

6.3 Basic Properties of and

133

(7)

6.4 Infinite sets without AC - countable sets

140

(11)

6.5 Uncountable sets and cardinal arithmetic, without AC

151

(12)

7 Ordered Sets

163

(39)

7.1 Introduction

163

(1)

7.2 Linearly ordered sets

164

(13)

7.3 Order arithmetic

177

(11)

7.4 Well-ordered sets

188

(14)

8 Ordinal Numbers

202

(61)

8.1 Introduction

202

(1)

8.2 Ordinal numbers

203

(15)

8.3 Beginning ordinal arithmetic

218

(13)

8.4 Ordinal arithmetic

231

(23)

8.5 The Ns

254

(9)

9 Set Theory with the Axiom of Choice

263

(18)

9.1 Introduction

263

(1)

9.2 The well-ordering principle

264

(3)

9.3 Cardinal arithmetic and the axiom of choice

267

(9)

9.4 The continuum hypothesis

276

(5)

Bibliography

281

(2)

Index

283


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