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Basic Set Theory 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 OSNOVY TEORII MNOESTV MCNMO, Moskva, 1999 Translated from the Russian by A. Shen 2000 Mathematics Subject Classification. Primary 03–01, 03Exx. Abstract. 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. Library of Congress Cataloging-in-Publication Data Vereshchagin, Nikolai Konstantinovich, 1958– [Osnovy teorii mnozhestv. English] Basic set theory / A. Shen, N. K. Vereshchagin. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 17) Authors’ names on t.p. of translation reversed from original. Includes bibliographical references and index. ISBN 0-8218-2731-6 (acid-free paper) 1. Set theory. I. Shen, A. (Alexander), 1958– II. Title. III. Series. QA248 .V4613 2002 511.322—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 passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294, USA. Requests can also be made by e-mail to [email protected]. c 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 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- theoretic topology). Usually these notions are briefly discussed in the opening chapters of textbooks on analysis, algebra, or topology, before passing to the main topic of the book. This is, however, unfortunate— the subject is sufficiently interesting, important, and simple to deserve 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 to a professional mathematician (who, on his/her way to vacations, wants to finally find out what is this transfinite indiction 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 to our teacher Vladimir Andreevich Uspensky, whose lectures, books, vii viii Preface and comments influenced us (and this book) perhaps even more 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 Bibliography [1] P. S. Aleksandrov, Introduction to set theory and general topology, “Nauka”, Moscow, 1977. (Russian) [2] N. Bourbaki, El´´ ements de Math´ematique XXII, Th´eorie des ensembles, Hermann, Paris, 1957. [3] G. Cantor, Works in set theory, Compiled by A. N. Kolmogorov, F. A. Medvedev, and A. P. Yushkevich, “Nauka”, Moscow, 1985. (Rus- sian)1 [4] P. J. Cohen, Set theory and the continuum hypothesis,Benjamin,New York, 1966. [5] A. A. Fraenkel and Y. Bar-Hillel, Foundations of set theory, Studies in Logic and the Foundations of Mathematics, North-Holland, Amster- dam, 1958. [6] Handbook of mathematical logic, Edited by Jon Barwise, with the coop- erationofH.J.Keisler,K.Kunen,Y.N.Moschovakis,andA.S.Troel- stra, Studies in Logic and the Foundations of Mathematics, Vol. 90, North-Holland, Amsterdam, 1977. [7] F. Hausdorff, Grundz¨uge der Mengenlehre, Veit, Leipzig, 1914. [8]T.J.Jech,Lectures on set theory with particular emphasis on the method of forcing, Springer-Verlag, Berlin, 1971. [9] W. Just and M. Weese, Discovering modern set theory, I. The basics, Amer. Math. Soc., Providence, RI, 1996. 1Editorial Note. This collection consists mostly of selected works translated from the complete collection (Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Reprint of the 1932 original, Springer-Verlag, Berlin–New York, 1980). More on the content of the Russian book can be found in Mathematical Reviews,MR87g:01062. 109 110 Bibliography [10] , Discovering modern set theory, II. Set-theoretic tools for every mathematician, Amer. Math. Soc., Providence, RI, 1997. [11] A. Kechris, Classical descriptive set theory, Springer-Verlag, New York, 1995. [12] K. Kuratowski and A. Mostowski, Set theory, North-Holland, Amster- dam, 1976. [13] Yu. Manin, A course in mathematical logic, Springer-Verlag, New York, 1991. [14] A. Mostowski, Constructible sets with applications, North-Holland, Amsterdam, 1969. [15] J. R. Shoenfield, Mathematical logic, Addison-Wesley, Reading, MA, 1967. Glossary Felix BERNSTEIN, Feb. 24, 1878, Halle (Germany) – Dec. 3, 1956, Zurich (Switzerland), 16, 20 F´elix Edouard´ Justin Emile´ BOREL, Jan. 7, 1871, Saint Affrique, Aveyron, Midi-Pyr´en´ees (France) – Feb. 3, 1956, Paris (France), 100 Luitzen Egbertus Jan BROUWER, Feb. 27, 1881, Overschie (now in Rot- terdam, Netherlands) – Dec. 2, 1966, Blaricum (Netherlands), 15 Cesare BURALI-FORTI, Aug. 13, 1861, Arezzo (Italy) – Jan. 21, 1931, Turin (Italy), 84 Georg Ferdinand Ludwig Philipp CANTOR, Mar. 3, 1845, St. Petersburg (Russia) – Jan. 6, 1918, Halle (Germany), 2, 15, 16, 20, 24–26, 29, 68 Paul Joseph COHEN, born Apr. 2, 1934, Long Branch, New Jersey (USA), 11 Julius Wilhelm Richard DEDEKIND, Oct. 6, 1831, Braunschweig (now Germany) – Feb. 12, 1916, Braunschweig (Germany), 13, 15 EUCLID of Alexandria, about 325 (?) BC – about 265 (?) BC, Alexandria (now Egypt), 29 Adolf Abraham Halevi FRAENKEL, Feb. 17, 1891, Munich (Germany) – Oct. 15, 1965, Jerusalem (Israel), 29, 85 Guido FUBINI, Jan. 19, 1879, Venice (Italy) – Jun. 6, 1943, New York (USA), 108 Galileo GALILEI, Feb. 15, 1564, Pisa (now Italy) – Jan. 8, 1642, Arcetri near Florence (now Italy), 17 111 112 Glossary Kurt GODEL,¨ Apr. 28, 1906, Br¨unn, Austria-Hungary (now Brno, Czech Republic) – Jan. 14, 1978, Princeton (USA), 11 Georg Karl Wilhelm HAMEL, Sep. 12, 1877, D¨uren, Rheinland (Ger- many) – Oct. 4, 1954, Berlin (Germany), 69, 76 Charles HERMITE, Dec. 24, 1822, Dieuze, Lorraine (France) – Jan. 14, 1901, Paris (France), 25 David HILBERT, Jan. 23, 1862, K¨onigsberg, Prussia (now Kaliningrad, Russia) – Feb. 14, 1943, G¨ottingen (Germany), 72 Julius KONIG,¨ 1849, Gy¨or (Hungary) – 1913, Budapest (Hungary), 40 Kazimierz KURATOWSKI, Feb. 2, 1896, Warsaw (Poland) – Jun. 18, 1980, Warsaw (Poland), 34 Carl Louis Ferdinand von LINDEMANN, Apr. 12, 1852, Hannover (now Germany) – Mar. 6, 1939, Munich (Germany), 25 Joseph LIOUVILLE, Mar. 24, 1809, Saint-Omer (France) – Sep. 8, 1882, Paris (France), 25 Nikolai Ivanovich LOBACHEVSKY , Dec. 1, 1792, Nizhnii Novgorod (Rus- sia) – Feb. 24, 1856, Kazan (Russia), 29 Sir Isaac NEWTON, Jan. 4, 1643, Woolsthorpe, Lincolnshire (England) – Mar. 31, 1727, London (England), 7 Giuseppe PEANO, Aug. 27, 1858, Cuneo, Piemonte (Italy) – Apr. 20, 1932, Turin (Italy), 16 Frank Plumpton RAMSEY, Feb. 22, 1903, Cambridge, Cambridgeshire (England) – Jan. 19, 1930, London (England), 42 Bertrand Arthur William RUSSELL, May 18, 1872, Ravenscroft, Trelleck, Monmouthshire (Wales, UK) – Feb. 2, 1970, Penrhyndeudraeth, Merioneth (Wales, UK), 28 Friedrich Wilhelm Karl Ernst SCHRODER,¨ Nov. 25, 1841, Mannhein (Germany) – Jun. 16, 1902, Karlsruhe (Germany), 20 John von NEUMANN, Dec. 28, 1903, Budapest (Hungary) – Feb. 8, 1957, Washington, D.C. (USA), 83, 84 Norbert WIENER, Nov. 26, 1894, Columbia, Missouri (USA) – Mar. 18, 1964, Stockholm (Sweden), 35 Ernst Friedrich Ferdinand ZERMELO, Jul.
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