About the BookAbout the AuthorTable of Contents

The book is based on lectures given by the authors to undergraduate students at Moscow
State University. It explains basic notions of “naive” set theory (cardinalities, ordered sets, transfinite induction, ordinals). The book can be read by undergraduate and graduate students and all those interested in basic notions of set theory. The book contains more than 100 problems of various degrees of difficulty.

A. Shen and N. K. Vereshchagin

Editorial Board
David Bressoud, Carl Pomerance, Robert Devaney, Hung-Hsi Wu

Preface vii
Chapter 1. Sets and Their Cardinalities 1
1. Sets 1
2. Cardinality 4
3. Equal cardinalities 7
4. Countable sets 9
5. Cantor–Bernstein Theorem 16
6. Cantor’s Theorem 24
7. Functions 30
8. Operations on cardinals 35

Chapter 2. Ordered Sets 41
1. Equivalence relations and orderings 41
2. Isomorphisms 47
3. Well-founded orderings 52
4. Well-ordered sets 56
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vi Contents
5. Transfinite induction 59
6. Zermelo’s Theorem 66
7. Transfinite induction and Hamel basis 69
8. Zorn’s Lemma and its application 74
9. Operations on cardinals revisited 78
10. Ordinals 83
11. Ordinal arithmetic 87
12. Recursive definitions and exponentiation 91
13. Application of ordinals 99

Bibliography 109
Glossary 111
Index 113