Set Theory: Techniques and Applications Curaçao 1995 and Barcelona 1996 Conferences

228 Pages·1998·10.42 MB·English

by  Maxim R. Burke (auth.), Carlos Augusto Di Prisco, Jean A. Larson, Joan Bagaria, A. R. D. Mathias (eds.)

Set Theory
Set Theory
Techniques and Applications
Curacao 1995 and Barcelona 1996 Conferences
edited by
Carlos Augusto Di Prisco
Instituto Yenezolano de Investigaciones Cientificas,
Caracas, Venezuela
Jean A. Larson
University of Florida.
Gainesville, Florida, U.S.A.
Joan Bagaria
Universitat de Barcelona.
Barcelona, Spain
and
A.R.D. Mathias
University of Wales,
Aberystwyth, Wales, U.K.
Springer-Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4978-0 ISBN 978-94-015-8988-8 (eBook)
DOI 10.1007/978-94-015-8988-8
Printed on acid-free paper
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© Springer Science+Business Media Dordrecht 1998
Originally published by Kluwer Academic Publishers in 1998.
Softcover reprint of the hardcover Ist edition 1998
No part of the materialprotectedby this copyrightnotice may be reproduced or
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TABLE OF CONTENTS
Preface vii
List of Participants ix
Articles
MAXIM R. BURKE
Forcing axioms . . . . . . . . . . . . . . . . 1
JAMES CUMMINGS
Large cardinal properties of small cardinals 23
CARL DARBY and RICHARD LAVER
Countable length Ramsey games . . . . . . 41
OMAR DE LA CRUZ and CARLOS A. 01 PRISCO
Weak forms of the axiom of choice and partitions of infinite sets 47
MARTIN GOLDSTERN
A taste of proper forcing . . 71
PIOTR KOSZMIDER
Applications of p-functions 83
Models as side conditions . 99
JEAN A. LARSON
An ordinal partition from a scale 109
A picaresque approach to set theory genealogy 127
A.R.D . MATHIAS
Recurrent points and hyperarithmetic sets. 157
E.C. MILNER and SAHARON SHELAH
A tree-arrowing graph . . . . . . . . . . . . 175
WILLIAM J . MITCHELL
A Hollow Shell: Covering Lemmas Without a Core 183
CARLOS H. MONTENEGRO
Partition properties for reals 199
ERNEST SCHIMMERLING
Combinatorial set theory and inner models 207
STEVO TODORCEVIC
Definable ideals and gaps in their quotients 213
PREFACE
This volume grew out of a pair of conferences both focused on techniques and
applications. The first conference, Combinatorial Set Theory and its Consequences,
was a Joint US-Venezuela Conference co-organized by Carlos A. Di Prisco and Jean
A. Larson. It was held June 26-30,1995 at the Avila Beach Hotel in Curacao in the
Netherlands Antilles. The second conference, Techniques and Applications of Set
Theory was co-organized by Joan Bagaria and Adrian Mathias. It was held June
10-14, 1996 at the Centre de Recerca Matematica (CRM) in Barcelona, Spain.
In his speech at the opening of the new CRM buildings, Professor Hilton
made a contrast between vertical and horizontal research. Vertical research, in
his metaphor, is the development of a branch of mathematics within itself, when
techniques become stronger, but where there is a concomitant danger of the sub-
ject becoming less accessible to outsiders. Horizontal research is the building of
bridges between these vertically developed branches of mathematics.
In the last twenty years, the interior techniques of Set Theory have developed
enormously: one may point to advances in the inner model program, the concept
of proper forcing, the axiom of determinacy, the set theory of the real line, and so
on. There are growing signs that the time is ripe for set theorists to seek to build
links with other areas .
One of the aims of the Barcelona meeting was to have a chance to learn some of
the techniques developed by Stevo Todorcevic, so two of the four short courses at
the Barcelona meeting were devoted to his work. The course given by Max Burke
focused on partition relations and forcing axioms, including several applications
to Topology and Measure Theory. Piotr Koszmider's course expounded two of the
most successful tools developed by Todorcevic, namely, p-functions, and forcing
with models as side conditions, and included some applications.
In their respective courses, Martin Goldstern gave an introduction to proper
forcing and James Cummings treated large-cardinal combinatorial properties of
small cardinals.
In this volume the reader will find expanded versions of the four courses. We
believe it provides a very good and accessible introduction to these beautiful, albeit
technically difficult subjects, and we hope they will be put to use to solve many
mathematical problems.
The conference in Curacao focused on combinatorial aspects of set theory. One
of its aims was to promote cooperation between researchers in this area of set
theory from different parts of the world. Another aim was to stimulate set theo-
retical research in Latin America. Ordinals appear in several papers . Carl Darby
and Richard Laver discuss games played on ordinals. Jean Larson uses short scales
to build interesting ordinal partitions. A.R.D. Mathias studies closure ordinals of
viii
iterations defined using an initial point and continuous function mapping a Polish
space into itself.
Partition theorems are another theme. Omar De la Cruz and Carlos A. Di
Prisco discuss weak forms of the axiom of choice and partitions of infinite sets.
Eric C. Milner and Saharon Shelah build a rich graph so that for every coloring
of its edges with two colors, there is an induced large tree all of whose edges
have the same color. Carlos Montenegro looks at a partition relation viewed as a
generalization of the Hales-Jewett Theorem.
Inner model theory has enhanced our understanding of possible cardinalities
and produced useful tools. William J. Mitchell makes a start on a program to
classify the failure of the Weak Covering Lemma at a Woodin cardinal ; Ernest
Schimmerling surveys a hierarchy of square principles extracted from core model
theory.
Stevo Todorcevic uses a covering property of definable sets of reals to analyse
definable gaps in definable ideals.
There is also an historical article by Jean Larson examining the mathematical
genealogy of the participants of the two conferences.
We thank Jorge Martinez, director of the the Caribbean Mathematics Founda-
tion, for his work as liason in Curacao and the support of his foundation. Addition-
ally we thank the National ScienceFoundation (NSF INT-9503676) and CONICIT
for their financial support of the conference in Curacao,
We thank the CRM director, Manuel Castellet, as wellas the CRM staff, Maria
Julia, Consol Roca and Joan Codina for their invaluable assistance. We also want
to thank the Generalitat de Catalunya (CmIT) and the Ministerio de Educacion
y Ciencia for their financial support.
Finally, we thank Arlene Williams of the University of Florida for her help in
preparing this document.
LIST OF PARTICIPANTS
Participants are listed with affiliations. The initials of Barcelona and Curacao
are used as superscripts Band C to indicate the conference(s) attended. Individuals
who have a second affiliation in parentheses were at the latter institution during
the conference.
c
OFELIA T. ALAS , University of Sao Paulo
B
DAVID ANTON Universitat de Barcelona
c,
H. R . ANTONIUS Anton de Kom University, Suriname
JOAN BAGARIAB,c, Universitat de Barcelona
c,
JAMES E. BAUMGARTNER Dartmouth College
B,
ROGER BOSCH Universidad de Oviedo
B,
MAXIM R. BURKE University of Prince Edward Island
MARfA CARRASCO Universidad Simon Bolivar
B
JAMES CUMMINGS , Carnegie Mellon University
PATRICK DEHORNOyB, Universite de Caen
c,
OMAR DE LA CRUZ University of Florida
(Instituto Venezolano de Investigaciones Cientificas)
CARLOS A. DI PRISCOB,c
Instituto Venezolano de Investigaciones Cientificas
RAIMON ELGUETA" , Universitat Politecnica de Catalunya
DAVID ESPERE", Universitat de Barcelona
MARTIN GOLDSTERN", Technische Universitat Wien
c
C. W . GORISSON , Anton de Kom University, Suriname
B,
VILLE HAKULINEN University of Helsinki
c,
JAMES M. HENLE Smith College
B
IGNASI JANE Universitat de Barcelona
c
ALEXANDER S. KECHRIS California Institute of Technology
c,
MENACHEM KOJMAN Ben Gurion University
and Carnegie Mellon University
B
PIOTR KOSZMIDER Auburn University
B
ADAM KRAWCZYK Warsaw University
c,
CLAUDE LAFLAMME University of Calgary
c,
JEAN A. LARSON University of Florida
x
c
RICHARD LAVER , University of Colorado
c
JEFFREY LEANING , University of Florida
c
JIMENA LLOPIS , Universidad Simon Bolivar
B
J ORDI LOPEZ , Universitat Autonoma de Barcelona
c
MENACHEM MAGIDOR , Hebrew University
c
M. VICTORIA MARSHALL , Universidad Catolica de Chile
B
A.R.D. MATHIAS , University of Wales at Aberyswyth,
(Centre de Recerca Matematica, Belleterra)
c
GISELA MENDEZ Universidad Central de Venezuela
c
ARNOLD W. MILLER , University of Wisconsin, Madison
c
ERIC C. MILNER , University of Calgary
c
WILLIAM J . MITCHELL , University of Florida
c
CARLOS MONTENEGRO , Universidad de Los Andes, Bogota
B
CHARLES J .G . MORGAN , Merton College, Oxford
B
JUAN CARLOS MARTINEZ , Universitat de Barcelona
c
JUI?ITH ROITMAN University of Kansas
c
ERNEST SCHIMMERLING , University of California, Irvine
(Massachusetts Institute of Technology)
c
ELiAS TAHHAN-BITTAR , Universidad de Los Andes, Trujillo
c
STEVO TODORCEVIC , University of Toronto
CARLOS E. UZCATEGuf, Universidad de Los Andes, Merida
B
PAULI VAISANEN , University of Helsinki
c
ANDRES VILLAVECES , Universidad Nacional de Colombia
(University of Wisconsin, Madison)
c
W . HUGH WOODIN , University of California, Berkeley
B
PIOTR ZAKRZEWSKI , Warsaw University
FORCING AXIOMS
MAXIM R. BURKEl
Department of Mathematics
University of Prince Edward Island
Charlottetown, P.E.I., NB C1A 4P3
Canada
Abstract. This paper surveys combinatorial forcing axioms together with combi-
natorial and topological consequences and information about their consistency.
These notes provide a survey of some forcing axioms which are consequences
of the proper forcing axiom. The notes are a modified and expanded version of
the author's lectures given at the workshop "Techniques and Applications of Set
Theory" held in June 1996 at the CRM in Barcelona, Spain. We thank S. Todor-
cevic for extensive discussions of the material. We thank Maria Julia of the CRM
for transcribing the author's transparencies into TeX. The level of detail varies
from section to section. Section 1.4 especially - in which we give a proof, based
on a technique in [5], of the preservation of the properness isomorphism condition
under short countable support iterations - is much more detailed than the rest
of the paper. It is imported from an unpublished note of the author's. In the rest
of the paper, we have added to the content of the transparencies some references,
a few proofs and some comments, but there are many details left to the reader .
For unexplained set theoretic notation, see [6J or [12] .
1. A property of ideals
The first axiom, R( ·) defined below, is a common generalization of several results
of S. Todorcevic. It is a special case of Lemma 1* (p. 152) of [14] and is easily seen
to be equivalent to Theorem 6A of [4] (see Remark 1.1.4 below). The statement is
a type of Ramsey theorem concerning certain types of partitions of the countable
subsets of a set S , hence the choice of the letter R.
1Research supported by NSERC. The author thanks the Centre de Recerca Matem-
atica for financial support during his stay in Barcelona and the Department of Mathe-
matics at the University of Wisconsin-Madison for its hospitality while this paper was
being written.
C.A. Di Prisco et al. (ed.), Set Theory, 1-21.
© 1998 KluwerAcademic Publishers.
2 MAXIM R. BURKE
After introducing appropriate definitions, we will give several applications of
R( ·), followed by a proof that it follows from the Proper Forcing Axiom.
1.1. DEFINITIONS
Definition 1.1.1. Let I be an ideal of subsets of a set S containing all singletons.
For the purposes of these notes, we will let R(I) denote the following statement:
R(I) : either: S = U{An I n < w} where, for each n < W , [An]W ~ t,
or: for every f : S X WI -t I there is an uncountable A ~ S
such that An f(x, 0:) is finite for every z E A, 0: < WI .
Let R(·) denote the statement that R(I) holds for every ideal I.
Remark 1.1.2. If I is generated by N1 sets, then we can let {f(o:, x) I 0: E WI }
be a generating family of I for each xES to see that R(I) implies
either: S = U{An In < w} where, for each n < W, [AnJw ~ I
or: 3A E [S]W\ \IF E I (IA n FI < w)
Remark 1.1.3. If, for each x E S, f(o:, x) does not depend on 0:, then we may
think of f as function from S into I . Then by thinning out the set A so that the
sets An f(x) for x E A, form a ~-system and by discarding the root we may
assume that the sets An f(x) for x E A, are pairwise disjoint . Consequently we
may assume that if x E f(x) for all xES, then (A n f(x) = {x}) for all x E A.
Remark 1.1.4. The following equivalent formulation of R(·) is Theorem 6A of
[4J. Let S ~ X x Y.
either: there is a sequence (Yn In < w) of subsets of Y such that Y \
(U{ Yn I n < w} is countable and for every countable B ~ Yn
there is a finite I ~ X such that B ~ S[I]
or: whenever T ~ X x Y has horizontal sections of size :5 N1 (and
h : Y -t X) there is an uncountable A ~ Y such that An S[{x}]
is finite for every x E T-1[A] (and (h(y),z) rt S for all distinct
Y,z E A) .
We leave to the interested reader the easy exercise of verifying the equivalence
of the definition of R(·) with the formulation given in Remark 1.1.4.
Variations on the statements in Remarks 1.1.2 and 1.1.3 above are studied in
[8]. It followsfrom the proof of Theorem 3.12 of [8] that in the presence of Martin's
Axiom, the conclusion in Remark 1.1.2 can be improved to say that one of the
following possibilities holds:
(a). S =U{An In < w }, where [An]W ~ I for all n < Wi
(b). S = U{An I n < w}, where IAn n FI < w for all n < wand F E I; or
(c). there are A, BE [S]W\ so that [A]W ~ I and IB n FI < w for all F E I .
1.2. SOME CONSEQUENCES OF R(·)
The following consequences of RO are taken from [10], [l1J, [12]. For pedagogical
reasons this list is far from minimal.