Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic settheoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here.

SECTION

PREFACE

1 THE AXIOM OF EXTENSION

2 THE AXIOM OF SPECIFICATION

3 UNORDERED PAIRS

4 UNIONS AND INTERSECTIONS

5 COMPLEMENTS AND POWERS

6 ORDERED PAIRS

7 RELATIONS

8 FUNCTIONS

9 FAMILIES

10 INVERSES AND COMPOSITES

11 NUMBERS

12 THE PEANO AXIOMS

13 ARITHMETIC

14 ORDER

15 THE AXIOM OF CHOICE

16 ZORN'S LEMMA

17 WELL ORDERING

18 TRANSFINITE RECURSION

19 ORDINAL NUMBERS

20 SETS OF ORDINAL NUMBERS

21 ORDINAL ARITHMETIC

22 THE SCHRODER-BERNSTEIN THEOREM

23 COUNTABLE SETS

24 CARDINAL ARITHMETIC

25 CARDINAL NUMBERS

INDEX