Elements of Set Theory

This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.

Undergraduate students interested in set theory.

Contents

Preface

List of Symbols

Chapter 1 Introduction

Baby Set Theory

Sets—An Informal View

Classes

Axiomatic Method

Notation

Historical Notes

Chapter 2 Axioms and Operations

Axioms

Arbitrary Unions and Intersections

Algebra of Sets

Epilogue

Review Exercises

Chapter 3 Relations and Functions

Ordered Pairs

Relations

n-Ary Relations

Functions

Infinite Cartesian Products

Equivalence Relations

Ordering Relations

Review Exercises

Chapter 4 Natural Numbers

Inductive Sets

Peano's Postulates

Recursion on ω

Arithmetic

Ordering on ω

Review Exercises

Chapter 5 Construction of the Real Numbers

Integers

Rational Numbers

Real Numbers

Summaries

Two

Chapter 6 Cardinal Numbers and the Axiom of Choice

Equinumerosity

Finite Sets

Cardinal Arithmetic

Ordering Cardinal Numbers

Axiom of Choice

Countable Sets

Arithmetic of Infinite Cardinals

Continuum Hypothesis

Chapter 7 Orderings and Ordinals

Partial Orderings

Well Orderings

Replacement Axioms

Epsilon-Images

Isomorphisms

Ordinal Numbers

Debts Paid

Rank

Chapter 8 Ordinals and Order Types

Transfinite Recursion Again

Alephs

Ordinal Operations

Isomorphism Types

Arithmetic of Order Types

Ordinal Arithmetic

Chapter 9 Special Topics

Well-Founded Relations

Natural Models

Cofinality

Appendix Notation, Logic, and Proofs

Selected References for Further Study

List of Axioms

Index