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Basic Set Theory
A. Shen N. K. Vereshchagin
STUDENT MATHEMAT ICAL L IBRARYVolume 17
Basic Set Theory
http://dx.doi.org/10.1090/stml/017
Basic Set TheoryA. ShenN. K.Vereshchagin
STUDENT MATHEMAT IC AL L IBRARYVolume 17
Editorial Board
David Bressoud, ChairRobert Devaney
Carl PomeranceHung-Hsi Wu
N. K. Verewagin, A. Xen�
OSNOVY TEORII MNO�ESTVMCNMO, 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 undergraduatestudents at Moscow State University. It explains basic notions of “naive” settheory (cardinalities, ordered sets, transfinite induction, ordinals). The book canbe read by undergraduate and graduate students and all those interested in basicnotions of set theory. The book contains more than 100 problems of variousdegrees 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 2002511.3′22—dc21 2002066533
Copying and reprinting. Individual readers of this publication, and nonprofitlibraries acting for them, are permitted to make fair use of the material, such as tocopy a chapter for use in teaching or research. Permission is granted to quote briefpassages from this publication in reviews, provided the customary acknowledgment ofthe source is given.
Republication, systematic copying, or multiple reproduction of any material in thispublication 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 reprint-permission@ams.org.
c© 2002 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept 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 guidelinesestablished 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 ver@mccme.ru or
shen@mccme.ru).
A. Shen, N. K. Vereshchagin
Bibliography
[1] P. S. Aleksandrov, Introduction to set theory and general topology,“Nauka”, Moscow, 1977. (Russian)
[2] N. Bourbaki, Elements de Mathematique XXII, Theorie 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, NewYork, 1966.
[5] A. A. Fraenkel and Y. Bar-Hillel, Foundations of set theory, Studies inLogic and the Foundations of Mathematics, North-Holland, Amster-dam, 1958.
[6] Handbook of mathematical logic, Edited by Jon Barwise, with the coop-eration of H. J. Keisler, K. Kunen, Y. N. Moschovakis, and A. S. Troel-stra, Studies in Logic and the Foundations of Mathematics, Vol. 90,North-Holland, Amsterdam, 1977.
[7] F. Hausdorff, Grundzuge der Mengenlehre, Veit, Leipzig, 1914.
[8] T. J. Jech, Lectures on set theory with particular emphasis on themethod 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 fromthe complete collection (Georg Cantor, Gesammelte Abhandlungen mathematischenund philosophischen Inhalts, Reprint of the 1932 original, Springer-Verlag, Berlin–NewYork, 1980). More on the content of the Russian book can be found in MathematicalReviews, MR 87g:01062.
109
110 Bibliography
[10] , Discovering modern set theory, II. Set-theoretic tools for everymathematician, Amer. Math. Soc., Providence, RI, 1997.
[11] A. Kechris, Classical descriptive set theory, Springer-Verlag, NewYork, 1995.
[12] K. Kuratowski and A. Mostowski, Set theory, North-Holland, Amster-dam, 1976.
[13] Yu. Manin, A course in mathematical logic, Springer-Verlag, NewYork, 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
Felix Edouard Justin Emile BOREL, Jan. 7, 1871, Saint Affrique, Aveyron,Midi-Pyrenees (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 (nowGermany) –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, Arcetrinear Florence (now Italy), 17
111
112 Glossary
Kurt GODEL, Apr. 28, 1906, Brunn, Austria-Hungary (now Brno, CzechRepublic) – Jan. 14, 1978, Princeton (USA), 11
Georg Karl Wilhelm HAMEL, Sep. 12, 1877, Duren, 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, Konigsberg, Prussia (now Kaliningrad,Russia) – Feb. 14, 1943, Gottingen (Germany), 72
Julius KONIG, 1849, Gyor (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 (nowGermany) –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
2X , 36
2N, cardinality, 24= (equality), 1A∗, 81
AB , 36An, 81Fσ , 103Gδ , 103P (U), 9, 30, 43, 49[B → A], 94DomF , 31ValF , 31ℵ0, 38ℵ1, 102
αβ , 93N∗, 105
Nk, 11, 46Q, 11R, 11, 25R, cardinality, 25Rn, cardinality, 16Z[x, . . . ], 38, 2(nk
), 6
∩, 1∪, 2idA, 31〈a, b〉, 30, 34Bα, 101c, 38π, transcendental number, 25 , 7⊂, 1�, 1{a, b, c}, 2, 34
a < b, 44e, transcendental number, 25f : A → B, 31
f−1(B), 32g ◦ f , 31k-subsets, 6|A|, 24
adding a countable set, 13addition, 87, 97
of cardinals, 35recursive definition, 92
adjacent elements, 51σ-algebra, 100algebraic numbers, 12, 25associativity, 87, 89automorphism, 49axiom
of choice, 10, 29, 34, 66of replacement, 93of separation, 30of union, 30power set, 30
axiomatic set theory, 85
base change, 91total, 99
basis, 69bijection, 33binary relation, 30, 41binary-rational number, 50binomial coefficient, 6binomial expansion, 6, 7Borel set, 100, 101bounded subset, 57branch, 105
113
114 Index
Cantor’s Theorem, 24, 27Cantor–Bernstein Theorem, 9, 18, 21,
39, 79Cantor–Schroder–Bernstein Theorem,
16cardinalities
addition of, 79comparison of, 16, 68multiplication of, 78, 81
cardinality, 4, 7, 14, 23, 30, 33, 79, 81cardinals
operations on, 35Cartesian product, 30, 40chain, 74characteristic function, 5characteristic sequence, 26closed set, 103
cardinality of, 26combinations, 6commutativity, 87, 89comparable elements, 43comparing cardinalities, 22complement of a set, 2componentwise ordering, 46composition, 31condensation point, 26congruent polyhedra, 72continuous fractions, 39continuum
cardinality, 14, 16Continuum Hypothesis, 26, 28, 38,
106, 107coordinatewise ordering, 43countable set, 9
dense set, 51descendant, 105descriptive set theory, 100diagonal construction, 24difference, 2, 88dimension, 82distributivity, 89division, 90domain, 31
elementgreatest, 47least, 47minimal, 47of a set, 1
elementary equivalence, 50empty set, 1equal sets, 1equidecomposable polyhedra, 72equivalence, 41
class, 41relation, 8
Euclid’s fifth postulate, 29
exponentiation, 91, 93, 97cardinals, 36
extensionality, 85
finite support, 94foundation, 85Fubini’s Theorem, 108function, 30
graph, 31partial, 31
graphtopologically sorted, 78
greatest element, 47greatest lower bound, 57
Hamel basis, 69, 75, 76, 82Hilbert’s Third Problem, 72
identity function, 31image, 32Inclusion-Exclusion Principle, 4induced order, 45induction, 52
transfinite, 59infinite formula, 106infinite sets, 9, 13infix notation, 30initial segment, 58, 64injection, 33intersection, 1inverse function, 33isolated point, 26isomorphic posets, 47isomorphism, 47
Koenig’s Theorem, 40
leaf, 106least upper bound, 57left inverse, 33lexicographical order, 44, 55limit element, 57limit ordinal, 84linear
combination, 69independence, 69order, 43, 77
lower bound, 57
mapping, 32maximal
element, 47maximum
point, 13monotone Boolean functions, 4multiplication, 89, 97
cardinals, 35recursive definition of, 92
Index 115
non-Euclidean geometry, 29nonlimit ordinal, 84null set, 1
one-to-one correspondence, 7, 16, 33open set, 103order
induced, 45lexicographical, 44linear (total), 43, 77partial, 77well-founded, 52, 54
order type, 48, 65, 83, 87ordered pair, 30, 34
Kuratowski’s definition, 34Wiener’s definition, 35
ordered set, 41, 43, 77ordering, 41, 43ordinal, 23, 65, 86
limit, 84nonlimit, 84von Neumann, 84
ordinals, 83addition of, 91comparing, 65comparison of, 83exponentiation of, 91, 93multiplication of, 92
paradoxbarber’s, 28Burali-Forti, 84liar’s, 28Russell’s, 28
partial order, 43, 77partially ordered set, 43, 77Peano’s curve, 16periodic fraction, 12pointwise ordering, 43polyhedron, 72poset, 43positional number system, 90positional system, 98power set, 9, 30, 43, 46, 49
cardinality, 27predecessor, 56prefix code, 9preimage, 32, 100preorder, 45product, 81, 87, 89
of cardinals, 35of linearly ordered sets, 46of ordered sets, 46of well-founded sets, 54
proper subset, 1
quotient, 90set, 42
Ramsey Theorem, 42range, 31rational numbers, 11real numbers, 11, 16, 25reflexive relation, 8reflexivity, 41regularity, 85relation, 30
ordering, 43preorder, 45
remainder, 90replacement, axiom of, 93right inverse, 33root, 105rooted tree, 105
separation, axiom of, 30set
cardinality of, 1empty, 1linearly independent, 69linearly ordered, 43partially ordered, 43totally ordered, 43
set theory, 85sets, 1
Borel, 100, 101closed, 103difference of, 2equal, 1intersection of, 1open, 103ordered, 41symmetric difference of, 2transitive, 86union of, 2well-founded, 104well-ordered, 56, 68
singleton, 27, 34son, 105strict ordering, 45subset, 1successor, 56sum, 87
of cardinalities, 79of cardinals, 35of posets, 45
superset, 1support, 94surjection, 33symmetric
difference, 2relation, 8
symmetry, 41
topologically sorted graph, 78total function, 31total order, 43, 77transcendental numbers, 25
116 Index
transfiniteinduction, 59, 69recursion, 59, 62, 69
theorem, 61transitive
relation, 8set, 86
transitivity, 41tree, 104
rooted, 105tree rank, 105
union, 2axiom of, 30
unit square, cardinality of, 15upper bound, 57, 74
value, 31vertex, 105
well-foundedordering, 52, 54set, 104
well-ordered sets, 68comparison of, 63
well-ordering, 56, 83
Zermelo’s Theorem, 66ZF, Zermelo–Fraenkel axiomatic
set theory, 29, 85ZFC, 29Zorn’s Lemma, 74, 99
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 varifoldgeometry, 2001
12 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysisII: Continuity and differentiation, 2001
11 Michael Mesterton-Gibbons, An introduction to game-theoreticmodelling, 2000
10 John Oprea, The mathematics of soap films: Explorations with Maple r©,2000
9 David E. Blair, Inversion theory and conformal mapping, 2000
8 Edward B. Burger, Exploring the number jungle: A journey intodiophantine analysis, 2000
7 Judy L. Walker, Codes and curves, 2000
6 Gerald Tenenbaum and Michel Mendes France, The prime numbersand their distribution, 2000
5 Alexander Mehlmann, The game’s afoot! Game theory in myth andparadox, 2000
4 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysisI: 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 contemporaryprobability, 1999
1 Charles Radin, Miles of tiles, 1999
Basic Set T
heory
Shen and Vereshchagin
The main notions of set theory (cardinals, ordinals, transfi-nite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment.
This book provides just that: a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional math-ematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn’s Lemma.
The text introduces all main subjects of “naive” (nonax- iomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discus-sions and proofs of the Cantor–Bernstein Theorem, Cantor’s diagonal method, Zorn’s Lemma, Zermelo’s Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.
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