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## Overview

A historical introduction presents a brief account of the growth of set theory, with special emphasis on problems that led to the development of the various systems of axiomatic set theory. Subsequent chapters explore classes and sets, functions, relations, partially ordered classes, and the axiom of choice. Other subjects include natural and cardinal numbers, finite and infinite sets, the arithmetic of ordinal numbers, transfinite recursion, and selected topics in the theory of ordinals and cardinals. This updated edition features new material by author Charles C. Pinter.

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## Product Details

ISBN-13: | 9780486497082 |
---|---|

Publisher: | Dover Publications |

Publication date: | 07/16/2014 |

Pages: | 256 |

Sales rank: | 923,357 |

Product dimensions: | 6.20(w) x 9.10(h) x 0.70(d) |

## About the Author

*Book of Abstract Algebra.*

## Table of Contents

Preface ix

Chapter 0 Historical Introduction

1 The background of set theory 1

2 The paradoxes 3

3 The axiomatic method 4

4 Axiomatic set theory 9

5 Objections to the axiomatic approach. Other proposals 13

6 Concluding remarks 18

Chapter 1 Classes and Sets

1 Building sentences 20

2 Building classes 24

3 The algebra of classes 29

4 Ordered pairs Cartesian products 33

5 Graphs 37

6 Generalized union and intersection 40

7 Sets 44

Chapter 2 Functions

1 Introduction 49

2 Fundamental concepts and definitions 50

3 Properties of composite functions and inverse functions 57

4 Direct images and inverse images under functions 62

5 Product of a family of classes 66

6 The axiom of replacement 70

Chapter 3 Relations

1 Introduction 71

2 Fundamental concepts and definitions 71

3 Equivalence relations and partitions 74

4 Pre-image, restriction and quotient of equivalence relations 79

5 Equivalence relations and functions 82

Chapter 4 Partially Ordered Classes

1 Fundamental concepts and definitions 86

2 Order preserving functions and isomorphism 89

3 Distinguished elements. Duality 93

4 Lattices 98

5 Fully ordered classes. Well-ordered classes 103

6 Isomorphism between well-ordered classes 106

Chapter 5 The Axiom of Choice and Related Principles

1 Introduction 110

2 The axiom of choice 153

3 An application of the axiom of choice 115

4 Maximal principles 118

5 The well-ordering theorem 120

6 Conclusion 121

Chapter 6 The Natural Numbers

1 Introduction 123

2 Elementary properties of the natural numbers 125

3 Finite recursion 128

4 Arithmetic of natural numbers 131

5 Concluding remarks 137

Chapter 7 Finite and Infinite Sets

1 Introduction 138

2 Equipotence of sets 142

3 Properties of infinite sets 144

4 Properties of denumerable sets 146

Chapter 8 Arithmetic of Cardinal Numbers

1 Introduction 150

2 Operations on cardinal numbers 152

3 Ordering of the cardinal numbers 156

4 Special properties of infinite cardinal numbers 160

5 Infinite sums and products of cardinal numbers 162

Chapter 9 Arithmetic of the Ordinal Numbers

1 Introduction 166

2 Operations on ordinal numbers 168

3 Ordering of the ordinal numbers 173

4 The alephs and the continuum hypothesis 180

5 Construction of the ordinals and cardinals 181

Chapter 10 Transfinite Recursion. Selected Topics in the Theory of Ordinals and Cardinals

1 Transfinite recursion 188

2 Properties of ordinal exponentiation 192

3 Normal form 197

4 Epsilon numbers 202

5 Inaccessible ordinals and cardinals 206

Chapter 11 Consistency and Independence in Set Theory

1 What is a set? 213

2 Models 219

3 Independence results in set theory 221

4 The question of models of set theory 222

5 Properties of the constructible universe 225

6 The Gödel Theorems 232

Bibliography 235

Index 237