The Philosophy of Set Theory An Historical

This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more. 1989 edition. Includes 32 figures.

Preface

xiii

Introduction: Invention or Discovery?

1

(5)

1 The Finite Universe

6

(26)

1 Finitism

6

(4)

2 Continuity and Infinity

10

(2)

3 Zeno's Paradoxes

12

(10)

4 The Universe and the Absolutely Infinite

22

(10)

2 Classes and Aristotelian Logic

32

(24)

1 Aristotelian Logic

33

(6)

2 Aristotle on Knowledge of Universals

39

(8)

3 Nominalism and Extensionalism

47

(3)

4 Toward an Algebra of Classes

50

(4)

5 Classical Finitism in Retreat?

54

(2)

3 Permutations, Combinations and Infinite Cardinalities

56

(12)

1 Finite Permutations and Combinations

57

(2)

2 Probabilities as Limits of Infinite Sequences

59

(3)

3 Infinite Cardinalities

62

(2)

4 The Finitist Response

64

(4)

4 Numbering the Continuum

68

(27)

1 The Algebraization of Geometry

69

(15)

2 The Arithmetization of Analysis

84

(4)

3 Toward Infinite Ordinal Numbers

88

(2)

4 Conflict with Classical Finitism

90

(5)

5 Cantor's Transfinite Paradise

95

(23)

1 Sets and Cardinal Numbers

96

(8)

2 Transfinite Ordinal Numbers

104

(3)

3 Ordinal Numbers and Cantor's Continuum Hypothesis

107

(4)

4 Order Types

111

(3)

5 Set Theoretic Paradoxes

114

(4)

6 Axiomatic Set Theory

118

(20)

1 Axiomatization

118

(3)

2 The ZF Axioms

121

(13)

3 Transfinite Numbers in ZF

134

(4)

7 Logical Objects and Logical Types

138

(37)

1 Frege, Logic and Arithmetic

138

(8)

2 Frege's Universe

146

(8)

3 Iterative Sets and Simple Types

154

(4)

4 Russell's Logicist Reduction

158

(17)

8 Independence Results and the Universe of Sets

175

(17)

1 Godel's Constructible Universe

175

(4)

2 Cardinals and Ordinals in Models

179

(4)

3 Inner Models

183

(2)

4 Generic Sets

185

(7)

9 Mathematical Structure - Construct and Reality

192

(32)

1 Appeals to Mathematical Intuition

192

(6)

2 Appeals to Mathematical Consequences

198

(3)

3 Descriptive Set Theory

201

(6)

4 Foundations and Superstructures

207

(17)

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The Philosophy of Set Theory An Historical Introduction to Cantor's Paradise

Summary

Table of Contents

Supplemental Materials