Algebras of Sets and Combinatorics

264 Pages·2002·13.698 MB·English

by  L. S. Grinblat

Translations of
MATHEMATICAL
MONOGRAPHS
Volume 214
Algebras of Sets
and Combinatorics
L. S. Grinblat
American Mathematical Society
Algebras of Sets
and Combinatorics
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 214
Algebras of Sets
and Combinatorics
L. S. Grinblat
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
AMS Subcommittee
Robert D. MacPherson
Grigorii A. Margulis
James D. Stasheff (Chair)
ASL Subcommittee Steifen Lempp (Chair)
IMS Subcommittee Mark I. Freidlin (Chair)
Л. Ш. Гринблат
АЛГЕБРЫ МНОЖЕСТВ И КОМБИНАТОРИКА
Translated from an original Russian manuscript by A. Stoyanovskii.
Translation edited by A. B. Sossinsky.
2000 Mathematics Subject Classification. Primary 03E05; Secondary 28A05, 54D35.
Library of Congress Cataloging-in-Publication Data
Grinblat, L. S. (Leonid S.), 1944-
[Algebry mnozhestv i kombinatorika. English]
Algebras of sets and combinatorics / L. S. Grinblat.
p. cm. — (Translations of mathematical monographs, ISSN 0065-9282 ; v. 214)
Includes bibliographical references and index.
ISBN 0-8218-2765-0 (alk. paper)
1. Combinatorial set theory. I. Title. II. Series.
QA248.G753 2002
511.3,22-dc21 2002074584
CIP
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].
© 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.
0 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
Chapter 1. Introduction 1
Chapter 2. Main Results 11
Chapter 3. The Main Idea 25
Chapter 4. Finite Sequences of Algebras (1). Proof of Theorems 2.1 and 2.2 39
Chapter 5. Countable Sequences of Algebras (1). Proof of Theorem 2.4 57
Chapter 6. Proof of the Gitik-Shelah Theorem, and More from Set Theory 71
Chapter 7. Proof of Theorems 1.17, 2.7, 2.8 83
Chapter 8. Theorems on Almost cr-Algebras. Proof of Theorem 2.9 93
Chapter 9. Finite Sequences of Algebras (2). The Function g(n) 109
Chapter 10. A Description of the Class of Functions 145
Chapter 11. The General Problem. Proof of Theorems 2.15 and 2.20 163
Chapter 12. Proof of Theorems 2.21(1,3), 2.24 175
Chapter 13. The Inverse Problem 179
Chapter 14. Finite Sequences of Algebras (3). Proof of Theorems 2.27, 2.31,
2.36, 2.38 181
Chapter 15. Preliminary Notions and Lemmas 197
Chapter 16. Finite Sequences of Algebras (4). Proof of Theorems 2.39(1,2),
2.45(1,2) 207
Chapter 17. Countable Sequences of Algebras (2). Proof of Theorems 2.29,
2.32, 2.46 229
Chapter 18. A Refinement of Theorems on cr-Algebras. Proof of Theorems
2.34, 2.44 235
Chapter 19. Semistructures and Structures of Sets. Proof of Theorem 2.48 239
Chapter 20. Final Comments. Generalization of Theorem 2.1 245
Appendix. On a Question of Grinblat by Saharon Shelah 247
CONTENTS
VI
251
Bibliography
255
Index
CHAPTER 1
Introduction
1.1. This monograph contains the results of [Gr] as well as many new results
I have obtained in recent years. The new results and the results from [Gr] are
so intertwined that it did not seem reasonable to publish all the new results in a
separate very large paper. In contrast to [Gr], in the present monograph the theory
is complete. Thus, the statements to which we refer without proofs and which we
use for proving the main results are widely known and are simple facts of modern
mathematics. This also means that all the main results entirely belong to “naive”
set theory (except for the results in the Appendix written by Saharon Shelah).
Let us start with some history. George Cantor, the creator of set theory, be­
gan studying sets in 1872. His starting point were papers on trigonometric series
inspired by the works of Riemann. In 1878 Cantor proposed the continuum hypoth­
esis (Hi = 2N°) which was placed first in Hilbert’s 1900 list of problems bequeathed
by him to the 20th century (see [Hi]). Breaking the chronology of our story, let
us mention that the solution of the first Hilbert problem turned out to be striking.
First, in the paper [Go] published in 1939, Godel proved the consistency of the
continuum hypothesis with the axioms of set theory, and in the early 60’s Paul Co­
hen developed a powerful method for constructing models, the forcing method, and
constructed a model of set theory in which Hi < 2**° (see [Col]).1 It became clear
that the persistent efforts of Cantor and other mathematicians to prove the con­
tinuum hypothesis were of no avail. The continuum hypothesis, like Euclid’s fifth
postulate, can be neither proved nor disproved. The last papers of Cantor date
back to 1895-1897; .they are mostly devoted to the theory of completely ordered
sets and to the calculus of ordinal numbers.2
In 1902 Lebesgue published his thesis [LI], which became the starting point of
modern measure theory. The notions of a Lebesgue measurable and nonmeasurable
set became fundamental in mathematics. New set theory problems arose. In the
book [L2], published in 1904, Lebesgue posed the following problem.
The Lebesgue Problem. Does there exist a nonnegative real function /i on
the set of all subsets of the interval X = (0,1) that satisfies the following conditions:
(a) if Mi and M2 are isometric sets, then /¿(Mi) = /¿(M2);
1 Although the reader is not assumed to be familiar with the forcing method (except in the
Appendix), it is related to our monograph. Some examples use results obtained by the forcing
method. The important theorem of Gitik-Shelah stated below was originally proved by the forcing
method. The infinite analog of the Lusin theorem obtained by Gitik and Shelah — a result close to
our subjects and discussed in Chapter 6 — was also proved by the forcing method, and up to now
its purely mathematical proof is not known. An interesting result cited in 5.28 and presented in
the Appendix was obtained by Saharon Shelah by the forcing method as the answer to a question
of the author. In 6.12 we discuss an interesting result of Shelah obtained using the forcing method.
2The above information about Cantor is taken from [Bol] and [Cofc].
1
2 1. INTRODUCTION
(b) if Mi,..., Mfc,... is an at most countable sequence of pairwise disjoint sets,
then M(U^fc) = X X M fc);
(c) n(X) = 1?
In 1905 Vitali [Vi] obtained the presently well-known negative solution of this
problem.
1.2. The following problem was posed by Banach.
The Banach Problem. What is the solution of the Lebesgue problem with
condition (a) replaced by the following: fJ>{{x}) = 0 for each point x G XI
In 1929 Banach and Kuratowski published a negative solution of this problem
under the assumption of the continuum hypothesis in [B-Ku].3 Probably around
that time, many specialists believed that, even without the continuum hypothesis
assumption, the Banach problem has a negative solution. Thus, in a well-known
book [Bi] by G. Birkhoff published in 1948, one of the problems is posed as follows:
prove the nonexistence of a nontrivial measure with countable additivity4 for all
subsets of the continuum, such that any point has measure 0, without assuming
the continuum hypothesis.
1.3. Remark. Here it is appropriate to recall the history of the study of other
problems whose starting point is the Lebesgue problem formulated in 1.1. (We use
the historical survey from [Bo2].) In 1914 Hausdorff proved in [Haul] that there
does not exist a nontrivial finitely additive measure that is well defined for all
subsets of the unit ball in the 3-dimensional Euclidean space and invariant with
respect to isometries. It was natural to investigate whether the same is true for
the line and the plane — the problem was solved brilliantly by Banach in 1923,
who showed in [B] that, on the contrary, it is just in these two cases that such a
measure does exist. In 1929 von Neumann showed in [Nel] that the reason for the
difference between the line and the plane on one hand, and the Euclidean space
of dimension greater than 2 on the other hand, has to do with the commutativity
of the group of rotations of the plane. Therefore the a-additivity condition in the
Lebesgue problem cannot be replaced by the finite additivity condition. Using the
notion of ultrafilter, one easily shows that the cr-additivity condition in the Banach
problem cannot be replaced by the finite additivity condition. Note that the idea of
the “averaging” method from [B] was used by Haar, who proved in 1933 (see [H])
the existence of an invariant measure for locally compact groups. This discovery
immediately allowed von Neumann to solve (see [Ne2]), for compact groups, the
fifth Hilbert problem on the characterization of Lie groups by purely topological
properties.
3The result of Banach-Kuratowski follows from Theorem 1.6. It also follows from Theo­
rem 6.8 (see 6.9).
4We consider only real nonnegative finite measure (i.e., the measure of a measurable set is
a real nonnegative finite number). A measure is called nontrivial if there exists a measurable
set whose measure is greater than 0. As usual, a countable additivity property is also called cr-
additivity, and a cr-additive measure is called, for short, a cr-measure. Sometimes we will consider
not only cr-additive measures, but also finitely additive measures which are not, in general, cr-
additive.
1. INTRODUCTION 3
1.4. In 1971, in the period started by Cohen’s great discovery, the paper [So]
by R. Solovay appeared with the following remarkable result:
Solovay’s Result. In a certain model there exists a a-additive extension of
the Lebesgue measure to all subsets of (0,1).
1.5. Let us return to the Banach-Kuratowski result mentioned in 1.2. After
this, S. Ulam proved in [Ul] the following important more general result:
Ulam’s Matrix. If a set X has cardinality Ni, then one can construct a ma­
trix of subsets of X,
(Ml Ml ... Ml ...\
Ml Ml ... Ml ...
Ml Ml ... Ml ...
v................
which has No tows and Ni columns, and
(a) Ml D M^ = 0 if a /?;
(b) # (* \ u m £ )< n0.s
1.6. The following theorem, due to Ulam, is an obvious corollary of the exis­
tence of Ulam’s matrix:
Theorem. Let p be a nontrivial a-measure on a set of cardinality Ni, and
let p({x}) = 0 for each one-point set {#}. Then there exist Ni pairwise disjoint
p-nonmeasurable sets.
1.7. Definition. A finitely additive measure p defined on a set X is said to
be two-valued if p({x}) = 0 for all x e X, p(X) = 1, and for any ¿¿-measurable
set M either p(M) = 0 or p(M) = 1. If p is also cr-additive, then it is said to be
a-two-valued.
1.8. The following Alouglu-Erdos theorem stated in [Er] follows rather easily
from Theorem 1.6.
The Alouglu-Erdos Theorem. Assume that a countable sequence of a-
two-valued measures on a set of cardinality Ni is given. Then there exists a set
that is nonmeasurable with respect to each of these measures.
It is clear that in the Alouglu-Erdos theorem the cr-two-valuedness condition is
inessential. The theorem is valid under the assumption that each of the measures
is nontrivial, cr-additive, and each one-point set has measure 0.
1.9. The Alouglu-Erdos theorem gives a partial answer to the following Ulam
problem stated in [Er] (see also [Er-Ha]).
The Ulam Problem. Find the minimal cardinal6 x* such that, for any fam­
ily (of cardinality less than x*) of a-two-valued measures defined on a set of car­
dinality Ni, there exists a set that is nonmeasurable with respect to each of the
measures in the family. 5 6
5In 6.4 we give a construction of the sets Ml. The symbol #(M) denotes the cardinality of
the set M. (In the Appendix the cardinality of a set M is denoted by \M\.)
6By a cardinal x we mean an ordinal number such that if a < x, then #(c*) < #{x). In
what follows, the cardinality of a set will be treated as a cardinal.
4 1. INTRODUCTION
Ulam proved that x* > No- By the Alouglu-Erdos theorem, x* > N1. In [S2]
Shelah has shown that in a certain model x* = N1. Note that in the Godel model
L, we have x* = N2.
1.10. In [Wo] Woodin improved Shelah’s result. He showed that in a certain
model there exists a a-two-valued measure ¡jl on a set X of cardinality N1 and a
family {Ma}a<u;i of ju-nonmeasurable sets such that
{a | #i(Aia \ M) = 0} ^ 0
for each M C X with ¿¿(M) ^ 0.
For each Ma let us define a a-two-valued measure on X as follows: =
0 if and only if fi(M fl Ma) = 0. It is evident that each subset of X is measurable
with respect to some measure fia. Therefore in the model considered by Woodin
we have x* = N1.
1.11. We will return to the history of the subject later. Now let us pass to
the main theme of our monograph.
Definition. By an algebra A on a set X we mean a nonempty system of
subsets of X possessing the following property: if Mi, M2 G A, then M\ U M2,
Mi \ M2 G A.
Unless specified otherwise, all algebras and measures are considered on some
abstract set X ^ 0. Unless the contrary follows from the context, by a set we
always mean a subset of X.
1.12. The following “general question” arises in connection with the Alouglu-
Erdos theorem and the Ulam problem. Assume we are given a family of measures
{¡i\}xeA (since the question is “general”, we do not specify whether they are a-
additive and so forth), and for each A G A there exists a /¿A-nonmeasurable set.
Does there exist a set nonmeasurable with respect to each of these measures? She­
lah’s and Woodin’s results are nontrivial examples giving a negative answer to our
question. But in these examples we deal with families such that #(A) = N1. And
what will be the answer if one considers families with #(A) < Ho? Let us ask our
question in a more general and, I believe, more natural form. Let us consider alge­
bras instead of measures. Thus, consider an at most countable family of algebras,
each of which is not equal to il(A’) (as usual, ty(X) denotes the set of all subsets
of AT). Does there exist a set which does not belong to any of these algebras? If we
are given two algebras A\ and A2, and each of them does not coincide with ^(X),
then it is easy to show that
A ^ A 2 ^ {X ).
1.13. However, if we take not two but three algebras each of which is not
equal to 93(AT), then it can happen that a set not belonging to any one of these
algebras does not exist.
Example. #(X) = 3 and X = {xi,X2,X3}. Define the algebras ^4i,^42)^3 by
listing their elements:
Ai = {0 , {xi}, {a?2) 2:3}, AT},
A2 = {0> {#2}) 1 > ^3}) -^0 >
^3 = {0 i{®3}»