Basic Set Theory PDF
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STUDENT MATHEMATICAL LIBRARY Volume 17 Basic Set Theory A. Shen N. K. Vereshchagin http://dx.doi.org/10.1090/stml/017 Basic Set Theory STUDENT MATHEMATICAL LIBRARY Volume 17 Basic Set Theory A. Shen N. K.Vereshchagin Editorial Board David Bressoud, Chair Carl Pomerance Robert Devaney Hung-Hsi Wu N. K. Verewagin, A. Xen(cid:2) OSNOVY TEORII MNO(cid:3)ESTV MCNMO, Moskva, 1999 Translated from the Russian by A. Shen 2000 Mathematics Subject Classification. Primary03–01, 03Exx. Abstract.Thebookisbasedonlecturesgivenbytheauthorstoundergraduate students at Moscow State University. It explains basic notions of “naive” set theory(cardinalities,orderedsets,transfiniteinduction,ordinals). Thebookcan bereadbyundergraduateandgraduatestudentsandallthoseinterestedinbasic notions of set theory. The book contains more than 100 problems of various degreesofdifficulty. Library of Congress Cataloging-in-Publication Data Vereshchagin,NikolaiKonstantinovich,1958– [Osnovyteoriimnozhestv. English] Basicsettheory/A.Shen,N.K.Vereshchagin. p.cm. —(Studentmathematicallibrary,ISSN1520-9121;v.17) Authors’namesont.p.oftranslationreversedfromoriginal. Includesbibliographicalreferencesandindex. ISBN0-8218-2731-6(acid-freepaper) 1.Settheory. I.Shen,A.(Alexander),1958– II.Title. III.Series. QA248.V4613 2002 511.3(cid:2)22—dc21 2002066533 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passagesfromthispublicationinreviews,providedthecustomaryacknowledgmentof thesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthis publicationispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requests for such permission should be addressed to the Acquisitions Department, AmericanMathematicalSociety,201CharlesStreet,Providence,RhodeIsland02904- 2294,[email protected]. (cid:2)c 2002bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 070605040302 Contents 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 v 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 Preface This book is based on notes from several undergraduate courses the authors offered for a number of years at the Department of Math- ematics and Mechanics of Moscow State University. (We hope to extend this series: the books “Calculi and Languages” and “Com- putable Functions” are in preparation.) The main notions of set theory (cardinals, ordinals, transfinite induction) are among those any professional mathematician should know (even if (s)he is not a specialist in mathematical logic or set- theoretictopology). Usuallythesenotionsarebrieflydiscussedinthe openingchaptersoftextbooksonanalysis,algebra,ortopology,before passingtothemaintopicofthebook. Thisis,however,unfortunate— thesubjectissufficientlyinteresting,important,andsimpletodeserve a leisurely treatment. It is such a leisurely exposition that we are trying to present here, having in mind a diversified audience: from an advanced high school student toa professional mathematician (who, on his/her way tovacations,wantstofinallyfindoutwhatisthistransfiniteindiction which is always replaced by Zorn’s Lemma). For deeper insight into set theory the reader can turn to other books (some of which are listed in references). We would like to use this opportunity to express deep gratitude toourteacherVladimirAndreevichUspensky,whoselectures,books, vii viii Preface and comments influencedus (andthisbook)perhaps evenmore than we realize. We are grateful to the AMS and Sergei Gelfand (who suggested to translate this book into English) for patience. We also thank Yuri Burman who helped a lot with the translation. Finally, we wish to thank all participants of our lectures and seminars and all readers of preliminary versions of this book. We would appreciate learning about all errors and typos in the book found by the readers (and sent by e-mail to [email protected] or [email protected]). A. Shen, N. K. Vereshchagin http://dx.doi.org/10.1090/stml/017/01 Chapter 1 Sets and Their Cardinalities 1. Sets Let us recall several operations on sets and notation for them: • A set consists ofelements. Notation: x∈M means that x is an element of a set M (belongs to M). • A set A is a subset of a set B (A ⊂ B) if each element of A is also an element of B. In this case B is called a superset of A. • Two sets A and B are equal (A = B) if they consist of the same elements (i.e., if A⊂B and B ⊂A). • If A is a subset of B and A (cid:4)= B, then A is called a proper subset of B (notation: A(cid:2)B). • The empty set ∅ (called also the null set) contains no ele- ments. It is a subset of any set. • The intersection A∩B of two sets A and B consists of all elements that belong both to A and to B: A∩B ={x|x∈A and x∈B}. 1
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