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

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c© 2002 by the American Mathematical Society. All rights reserved.The American Mathematical Society retains all rightsexcept those granted to the United States Government.

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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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