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 MNO ESTV 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. 27, 1871, Berlin (Germany) – May 21, 1953, Freiburg im Breisgau (Germany), 29, 66, 69, 85 Max ZORN, Jun. 6, 1906, Hamburg (Germany) – March 9, 1993, Bloom- ington, Indiana (USA), 74, 99 Index
2U ,9 a
113 114 Index
Cantor’s Theorem, 24, 27 exponentiation, 91, 93, 97 Cantor–Bernstein Theorem, 9, 18, 21, cardinals, 36 39, 79 extensionality, 85 Cantor–Schr¨oder–Bernstein Theorem, 16 finite support, 94 cardinalities foundation, 85 addition of, 79 Fubini’s Theorem, 108 comparison of, 16, 68 function, 30 multiplication of, 78, 81 graph, 31 cardinality, 4, 7, 14, 23, 30, 33, 79, 81 partial, 31 cardinals operations on, 35 graph Cartesian product, 30, 40 topologically sorted, 78 chain, 74 greatest element, 47 characteristic function, 5 greatest lower bound, 57 characteristic sequence, 26 closed set, 103 Hamel basis, 69, 75, 76, 82 cardinality of, 26 Hilbert’s Third Problem, 72 combinations, 6 commutativity, 87, 89 identity function, 31 comparable elements, 43 image, 32 comparing cardinalities, 22 Inclusion-Exclusion Principle, 4 complement of a set, 2 induced order, 45 componentwise ordering, 46 induction, 52 composition, 31 transfinite, 59 condensation point, 26 infinite formula, 106 congruent polyhedra, 72 infinite sets, 9, 13 continuous fractions, 39 infix notation, 30 continuum initial segment, 58, 64 cardinality, 14, 16 injection, 33 Continuum Hypothesis, 26, 28, 38, intersection, 1 106, 107 inverse function, 33 coordinatewise ordering, 43 isolated point, 26 countable set, 9 isomorphic posets, 47 isomorphism, 47 dense set, 51 descendant, 105 Koenig’s Theorem, 40 descriptive set theory, 100 diagonal construction, 24 leaf, 106 difference, 2, 88 least upper bound, 57 dimension, 82 left inverse, 33 distributivity, 89 lexicographical order, 44, 55 division, 90 limit element, 57 domain, 31 limit ordinal, 84 linear combination, 69 element independence, 69 greatest, 47 order, 43, 77 least, 47 lower bound, 57 minimal, 47 of a set, 1 mapping, 32 elementary equivalence, 50 maximal empty set, 1 element, 47 equal sets, 1 maximum equidecomposable polyhedra, 72 point, 13 equivalence, 41 monotone Boolean functions, 4 class, 41 multiplication, 89, 97 relation, 8 cardinals, 35 Euclid’s fifth postulate, 29 recursive definition of, 92 Index 115 non-Euclidean geometry, 29 Ramsey Theorem, 42 nonlimit ordinal, 84 range, 31 null set, 1 rational numbers, 11 real numbers, 11, 16, 25 reflexive relation, 8 one-to-one correspondence, 7, 16, 33 reflexivity, 41 open set, 103 regularity, 85 order relation, 30 induced, 45 ordering, 43 lexicographical, 44 preorder, 45 linear (total), 43, 77 remainder, 90 partial, 77 replacement, axiom of, 93 well-founded, 52, 54 right inverse, 33 order type, 48, 65, 83, 87 root, 105 ordered pair, 30, 34 rooted tree, 105 Kuratowski’s definition, 34 Wiener’s definition, 35 ordered set, 41, 43, 77 separation, axiom of, 30 ordering, 41, 43 set ordinal, 23, 65, 86 cardinality of, 1 limit, 84 empty, 1 nonlimit, 84 linearly independent, 69 von Neumann, 84 linearly ordered, 43 ordinals, 83 partially ordered, 43 addition of, 91 totally ordered, 43 comparing, 65 set theory, 85 comparison of, 83 sets, 1 exponentiation of, 91, 93 Borel, 100, 101 multiplication of, 92 closed, 103 difference of, 2 equal, 1 paradox intersection of, 1 barber’s, 28 open, 103 Burali-Forti, 84 ordered, 41 liar’s, 28 symmetric difference of, 2 Russell’s, 28 transitive, 86 partial order, 43, 77 union of, 2 partially ordered set, 43, 77 well-founded, 104 Peano’s curve, 16 well-ordered, 56, 68 periodic fraction, 12 singleton, 27, 34 pointwise ordering, 43 son, 105 polyhedron, 72 strict ordering, 45 poset, 43 subset, 1 positional number system, 90 successor, 56 positional system, 98 sum, 87 power set, 9, 30, 43, 46, 49 of cardinalities, 79 cardinality, 27 of cardinals, 35 predecessor, 56 of posets, 45 prefix code, 9 superset, 1 preimage, 32, 100 support, 94 preorder, 45 surjection, 33 product, 81, 87, 89 symmetric of cardinals, 35 difference, 2 of linearly ordered sets, 46 relation, 8 of ordered sets, 46 symmetry, 41 of well-founded sets, 54 proper subset, 1 topologically sorted graph, 78 total function, 31 quotient, 90 total order, 43, 77 set, 42 transcendental numbers, 25 116 Index transfinite value, 31 induction, 59, 69 vertex, 105 recursion, 59, 62, 69 theorem, 61 well-founded transitive ordering, 52, 54 relation, 8 set, 104 set, 86 well-ordered sets, 68 transitivity, 41 comparison of, 63 tree, 104 well-ordering, 56, 83 rooted, 105 tree rank, 105 Zermelo’s Theorem, 66 ZF, Zermelo–Fraenkel axiomatic union, 2 set theory, 29, 85 axiom of, 30 ZFC, 29 unit square, cardinality of, 15 Zorn’s Lemma, 74, 99 upper bound, 57, 74 Titles in This Series
17 A. Shen and N. K. Vereshchagin, Basic set theory, 2002 16 Wolfgang Kuhnel, ¨ Differential geometry: curves—surfaces—manifolds, 2002 15 Gerd Fischer, Plane algebraic curves, 2001 14 V. A. Vassiliev, Introduction to topology, 2001 13 Frederick J. Almgren, Jr., Plateau’s problem: An invitation to varifold geometry, 2001 12 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis II: Continuity and differentiation, 2001 11 Michael Mesterton-Gibbons, An introduction to game-theoretic modelling, 2000 10 John Oprea, The mathematics of soap films: Explorations with Mapler , 2000 9 David E. Blair, Inversion theory and conformal mapping, 2000 8 Edward B. Burger, Exploring the number jungle: A journey into diophantine analysis, 2000 7 Judy L. Walker, Codes and curves, 2000 6 G´erald Tenenbaum and Michel Mend`esFrance, The prime numbers and their distribution, 2000 5 Alexander Mehlmann, The game’s afoot! Game theory in myth and paradox, 2000 4 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis I: Real numbers, sequences and series, 2000 3 Roger Knobel, An introduction to the mathematical theory of waves, 2000 2 Gregory F. Lawler and Lester N. Coyle, Lectures on contemporary probability, 1999 1 Charles Radin, Miles of tiles, 1999
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The main notions of set theory (cardinals, ordinals, transfi- of set theory (cardinals, ordinals, The main notions not fundamental to all mathematicians, nite induction) are or set- specialize in mathematical logic only to those who a given Basic set theory is generally theoretic topology. or topology, in courses on analysis, algebra, brief overview interesting, and though it is sufficiently important, even treatment. leisurely own simple to merit its for a that: a leisurely exposition just This book provides for a broad range of audience. It is suitable diversified math- students to professional readers, from undergraduate out what transfinite to finally find ematicians who want replaced by Zorn’s induction is and why it is always Lemma. (nonax- of “naive” introduces all main subjects The text cardinalities, ordered and iomatic) set theory: functions, and its applications, well-ordered sets, transfinite induction Included are discus- ordinals, and operations on ordinals. Theorem, sions and proofs of the Cantor–Bernstein Lemma, Zermelo’s diagonal method, Zorn’s Cantor’s 150 problems, the over With Theorem, and Hamel bases. introduction to the book is a complete and accessible subject.