Chapter 1 INTRODUCTION .
Baby Set Theory 1
Sets--An Informal View
Classes 10
Axiomatic Method 10
Notation 13
Historical Notes 14
Chapter 2 AXIOMS AND OPERATIONS
Axioms 17
Arbitrary Unions and Intersections 23
Algebra of Sets 27
Epilogue 33
Review Exercises 33
Chapter 3 RELATIONS AND FUNCTIONS
Ordered Pairs 35
Relations 39
n-Ary Relations 41
Functions 42
Infinite Cartesian Products 54
Equivalence Relations 55
.Ordering Relations 62
Review Exercises 64
Chapter 4 NATURAL NUMBERS
Inductive Sets 67
Peano's Postulates 70
Recursion on w 73
Arithmetic 79
Ordering on w 83
Review Exercises 88
Chapter 5 CONSTRUCTION OF THE REAL NUMBERS ..
Integers 90
Rational Numbers 101
Real Numbers 111
Summaries 121
Two 123
Chapter 6 CARDINAL NUMBERS AND THE AXIOM
OF CHOICE
Equinumerosity 128
Finite Sets 133
Cardinal Arithmetic 138
Ordering Cardinal Numbers 145
Axiom of Choice 151
Countable Sets 159
Arithmetic of Infinite Cardinals 162
Continuum Hypothesis 165
Chapter 7 ORDERINGS AND ORDINALS
Partial Orderings 167
Well Orderings 172
Replacement Axioms 179
Epsilon-lmages 182
lsomorphisms 184
Ordinal Numbers 187
Debts Paid 195
Rank 200
Chapter 8 ORDINALS AND ORDER TYPES
Transfinite Recursion Again 209
Alephs 212
Ordinal Operations 215
Isomorphism Types 220
Arithmetic of Order Types 222
Ordinal Arithmetic 227
Chapter 9 SPECIAL TOPICS
Well-Founded Relations 241
Natural Models 249
Cofinality 257
Appendix NOTATION, LOGIC, AND PROOFS
Selected References for Further Study
List of Axioms
Index ...
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is a 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 intriguing results about 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. .
The author of a book always has a preferred manner for using the book: A reader should simply study it from beginning to end. But in practice, the users of a book have their own goals. I have tried to build into the present book enough flexibility to accommodate a variety of goals.The axiomatic material in the text is marked by a stripe in the margin. The purpose of the stripe is to allow a user to deemphasize the axiomatic material, or even to omit it entirely.
A course in axiomatic set theory might reasonably cover the first six or seven chapters, omitting Chapter 5. This is the amount of set theory that everyone with an interest in matters mathematical should know. Those with a special interest in set theory itself are encouraged to continue to the end of the book (and beyond). A very different sort of course might emphasizethe set-theoretic construction of the number systems. This course might cover the first five chapters, devoting only as much attention to the axiomatic material as desired. The book presupposes no specific background. It does give real proofs. The first difficult proof is not met until part way through Chapter 4. ..
The hierarchical view of sets, constructed by transfinite iteration of the power set operation, is adopted from the start. The axiom of regularity is not added until it can be proved to be equivalent to the assertion that every set has a rank.
The exercises are placed at the end of each (or nearly each) section. In addition, Chapters 2, :3, and 4 have "Review Exercises" at the ends of the chapters. These are comparatively straightforward exercises for the reader wishing additional review of the material. There are, in all, close to 300 exercises.
There is a brief appendix dealing with some topics from logic, such as truth tables and quantifiers. This appendix also contains an example of how one might discover a proof.
At the end of this text there is an annotated list of books recommended for further study. In fact it includes diverse books for several further studies in a variety of directions. Those wishing to track down the source of particular results or historical points are referred to the books on the list that provide specific citations.
There are two stylistic matters that require mention. The end of a proof is marked by a reversed turnstile (-I). This device is due to C. C. Chang and H.J. Keisler. In definitions, I generally pass up the traditionally correct "if" in favor of the logically correct "iff" (meaning "if and only if").
Two preliminary editions of the text have been used in my courses at UCLA. I would be pleased to receive comments and corrections from further users of the book. ..
