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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
Preface viiChapter 1. Sets and Their Cardinalities 11. Sets 12. Cardinality 43. Equal cardinalities 74. Countable sets 95. Cantor–Bernstein Theorem 166. Cantor’s Theorem 247. Functions 308. Operations on cardinals 35Chapter 2. Ordered Sets 411. Equivalence relations and orderings 412. Isomorphisms 473. Well-founded orderings 524. Well-ordered sets 56vvi Contents5. Transfinite induction 596. Zermelo’s Theorem 667. Transfinite induction and Hamel basis 698. Zorn’s Lemma and its application 749. Operations on cardinals revisited 7810. Ordinals 8311. Ordinal arithmetic 8712. Recursive definitions and exponentiation 9113. Application of ordinals 99Bibliography 109Glossary 111Index 113