Problems and Theorems in Classical Set Theory

491 Pages·2006·2.434 MB·English

by  Peter Komjath, Vilmos Totik

Problem Books in Mathematics
Edited by P. Winkler
Péter Komjáth and Vilmos Totik
Problems and Theorems
in Classical Set Theory
Péter Komjáth Vilmos Totik
Department of Computer Science Department of Mathematics
Eotvos Lorand University,Budapest University of South Florida
Budapest 1117 Tampa,FL 33620
Hungary USA
and
Series Editor: Bolyai Institute
Peter Winkler University of Szeged
Department of Mathematics Szeged
Dartmouth College Hungary
Hanover,NH 03755-3551 6720
[email protected] [email protected]
Mathematics Subject Classification (2000):03Exx,05-xx,11Bxx
Library ofCongress Control Number:2005938489
ISBN-10:0-387-30293-X
ISBN-13:978-0387-30293-5
Printed on acid-free paper.
© (2006) Springer Science+Business Media,LLC
All rights reserved.This work may not be translated or copied in whole or in part without
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Dedicated to Andr´as Hajnal
and to the memory of
Paul Erdo˝s and G´eza Fodor
Contents
Preface ..................................................... xi
Part I Problems
1 Operations on sets......................................... 3
2 Countability ............................................... 9
3 Equivalence................................................ 13
4 Continuum ................................................ 15
5 Sets of reals and real functions ............................ 19
6 Ordered sets............................................... 23
7 Order types ............................................... 33
8 Ordinals ................................................... 37
9 Ordinal arithmetic......................................... 43
10 Cardinals .................................................. 51
11 Partially ordered sets...................................... 55
12 Transfinite enumeration ................................... 59
13 Euclidean spaces........................................... 63
14 Zorn’s lemma.............................................. 65
viii Contents
15 Hamel bases ............................................... 67
16 The continuum hypothesis................................. 71
17 Ultrafilters on ω ........................................... 75
18 Families of sets ............................................ 79
19 The Banach–Tarski paradox ............................... 81
20 Stationary sets in ω1....................................... 85
21 Stationary sets in larger cardinals ......................... 89
22 Canonical functions........................................ 93
23 Infinite graphs............................................. 95
24 Partition relations .........................................101
25 Δ-systems .................................................107
26 Set mappings ..............................................109
27 Trees ......................................................111
28 The measure problem .....................................117
29 Stationary sets in [λ]<κ ....................................123
30 The axiom of choice .......................................127
31 Well-founded sets and the axiom of foundation ............129
Part II Solutions
1 Operations on sets.........................................135
2 Countability ...............................................147
3 Equivalence................................................159
4 Continuum ................................................163
5 Sets of reals and real functions ............................173
6 Ordered sets...............................................185
Contents ix
7 Order types ...............................................213
8 Ordinals ...................................................223
9 Ordinal arithmetic.........................................237
10 Cardinals ..................................................265
11 Partially ordered sets......................................275
12 Transfinite enumeration ...................................285
13 Euclidean spaces...........................................299
14 Zorn’s lemma..............................................309
15 Hamel bases ...............................................317
16 The continuum hypothesis.................................327
17 Ultrafilters on ω ...........................................341
18 Families of sets ............................................351
19 The Banach–Tarski paradox ...............................359
20 Stationary sets in ω1.......................................369
21 Stationary sets in larger cardinals .........................377
22 Canonical functions........................................385
23 Infinite graphs.............................................389
24 Partition relations .........................................405
25 Δ-systems .................................................421
26 Set mappings ..............................................427
27 Trees ......................................................433
28 The measure problem .....................................453
29 Stationary sets in [λ]<κ ....................................463
30 The axiom of choice .......................................471
31 Well-founded sets and the axiom of foundation ............481
x Contents
Part III Appendix
1 Glossary of Concepts ......................................493
2 Glossary of Symbols .......................................507
3 Index......................................................509
Preface
Althoughthefirstdecadesofthe20thcenturysawsomestrongdebatesonset
theory and the foundation of mathematics, afterwards set theory has turned
into a solid branch of mathematics, indeed, so solid, that it serves as the
foundation of the whole building of mathematics. Later generations, honest
to Hilbert’s dictum, “No one can chase us out of the paradise that Cantor
has created for us” proved countless deep and interesting theorems and also
applied the methods of set theory to various problems in algebra, topology,
infinitary combinatorics, and real analysis.
The invention of forcing produced a powerful, technically sophisticated
tool for solving unsolvable problems. Still, most results of the pre-Cohen era
can be digested with just the knowledge of a commonsense introduction to
the topic. And it is a worthy effort, here we refer not just to usefulness, but,
first and foremost, to mathematical beauty.
Inthisvolumeweofferacollectionofvariousproblemsinsettheory.Most
of classical set theory is covered, classical in the sense that independence
methods are not used, but classical also in the sense that most results come
fromtheperiod,say,1920–1970.Manyproblemsarealsorelatedtootherfields
of mathematics such as algebra, combinatorics, topology, and real analysis.
We do not concentrate on the axiomatic framework, although some as-
pects, such as the axiom of foundation or the roˆle of the axiom of choice, are
elaborated.
There are no drill exercises, and only a handful can be solved with just
understanding the definitions. Most problems require work, wit, and inspira-
tion. Some problems are definitely challenging, actually, several of them are
published results.
We have tried to compose the sequence of problems in a way that earlier
problems help in the solution of later ones. The same applies to the sequence
of chapters. There are a few exceptions (using transfinite methods before
their discussion)—those problems are separated at the end of the individual
chapters by a line of asterisks.
We have tried to trace the origin of the problems and then to give proper
reference at the end of the solution. However, as is the case with any other
mathematical discipline, many problems are folklore and tracing their origin
was impossible.
The reference to a problem is of the form “Problem x.y” where x denotes
the chapter number and y the problem number within Chapter x. However,
within Chapter x we omit the chapter number, so in that case the reference
is simply “Problem y”.
For the convenience of the reader we have collected into an appendix all
the basic concepts and notations used throughout the book.
AcknowledgementsWethankP´eterVarju´andGergelyAmbrusfortheir
carefulreadingofthemanuscriptandtheirsuggestionstoimprovethepresen-
xii Preface
tation.Collectingandwritinguptheproblemstookmanyyears,duringwhich
the authors have been funded by various grants from the Hungarian National
Science Foundation for Basic Research and from the National Science Foun-
dation (latest grants are OTKA T046991, T049448 and NSF DMS-040650).
We hope the readers will find as much enjoyment in solving some of the
problems as we have found in writing them up.
P´eter Komj´ath and Vilmos Totik
Budapest and Szeged-Tampa, July 2005