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Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Survey Papers J. Bagaria, N. Castells and P. Larson An Ω-logic Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
M. Bekkali and D. Zhani Upper Semi-lattice Algebras and Combinatorics . . . . . . . . . . . . . . . . . . . . . .
29
R. Bosch Small Definably-large Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
A. E. Caicedo Real-valued Measurable Cardinals and Well-orderings of the Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
A. Marcone Complexity of Sets and Binary Relations in Continuum Theory: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 A.R.D. Mathias Weak Systems of Gandy, Jensen and Devlin . . . . . . . . . . . . . . . . . . . . . . . . .
149
B. Tsaban Some New Directions in Infinite-combinatorial Topology . . . . . . . . . . . . . 225 Research Papers T. Banakh and A. Blass The Number of Near-Coherence Classes of Ultrafilters is Either Finite or 2c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 S.-D. Friedman Stable Axioms of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
275
S.-D. Friedman Forcing with Finite Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 G. Hjorth Subgroups of Abelian Polish Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
vi
Contents
P. Koepke and P. Welch On the Strength of Mutual Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 P. Matet Part(κ, λ) and Part∗ (κ, λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321
C. Morgan Local Connectedness and Distance Functions . . . . . . . . . . . . . . . . . . . . . . . .
345
R. Schindler Bounded Martin’s Maximum and Strong Cardinals . . . . . . . . . . . . . . . . . .
401
Foreword This is a collection of articles on set theory written by some of the participants in the Research Programme on Set Theory and its Applications that took place at the Centre de Recerca Matem` atica (CRM) in Bellaterra (Barcelona). The Programme run from September 2003 to July 2004 and included an international conference on set theory in September 2003, an advanced course on Ramsey methods in analysis∗ in January 2004, and a joint CRM-ICREA workshop on the foundations of set theory in June 2004, the latter held in Barcelona. A total of 33 short and long term visitors from 15 countries participated in the Programme. This volume consists of two parts, the first containing survey papers on some of the mainstream areas of set theory, and the second containing original research papers. All of them are authored by visitors who took part in the set theory Programme or by participants in the Programme’s activities. The survey papers cover topics as Omega-logic, applications of set theory to lattice theory and Boolean algebras, real-valued measurable cardinals, complexity of sets and relations in continuum theory, weak subsystems of axiomatic set theory, definable versions of large cardinals, and selection theory for open covers of topological spaces. As for the research papers, they range from topics such as the number of near-coherence classes of ultrafilters, the consistency strength of bounded forcing axioms, Pκ (λ) combinatorics, some applications of morasses, subgroups of Abelian Polish groups, adding club subsets of ω2 with finite conditions, the consistency strength of mutual stationarity, and new axioms of set theory. We would like to thank all participants in the Programme and its related activities for their effort in making these very successful ventures. We also want to thank the CRM Director and its staff, as well as all the funding institutions, for making the Programme possible. Joan Bagaria Stevo Todorcevic Editors ∗ The
volume: Spiros A. Argyros and Stevo Todorcevic, Ramsey Methods in Analysis. Advanced Courses in Mathematics CRM Barcelona. Birkh¨ auser 2005, contains the notes of the course in an expanded form.
Set Theory Trends in Mathematics, 1–28 c 2006 Birkh¨ auser Verlag Basel/Switzerland
An Ω-logic Primer Joan Bagaria, Neus Castells and Paul Larson Abstract. In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ω-logic and the Ω-conjecture. Keywords. Ω-logic, Woodin cardinals, A-closed sets, universally Baire sets, Ω-conjecture.
Introduction One family of results in modern set theory, called absoluteness results, shows that the existence of certain large cardinals implies that the truth values of certain sentences cannot be changed by forcing1 . Another family of results shows that large cardinals imply that certain definable sets of reals satisfy certain regularity properties, which in turn implies the existence of models satisfying other large cardinal properties. Results of the first type suggest a logic in which statements are said to be valid if they hold in every forcing extension. With some technical modifications, this is Woodin’s Ω-logic, which first appeared in [12]. Results of the second type suggest that there should be a sort of internal characterization of validity in Ω-logic. Woodin has proposed such a characterization, and the conjecture that it succeeds is called the Ω-conjecture. Several expository papers on Ω-logic and the Ω-conjecture have been published [1, 13, 14, 15, 16, 17]. Here we briefly discuss The first author was partially supported by the research projects BFM2002-03236 of the Ministerio de Ciencia y Tecnolog´ıa, and 2002SGR 00126 of the Generalitat de Catalunya. The third author was partially supported by NSF Grant DMS-0401603. This paper was written during the third author’s stay at the Centre de Recerca Matem` atica (CRM), whose support under a Mobility Fellowship of the Ministerio de Educaci´ on, Cultura y Deportes is gratefully acknowledged. It was finally completed during the first and third authors’ stay at the Institute for Mathematical Sciences, National University of Singapore, in July 2005. 1 Throughout this paper, by “forcing” we mean “set forcing”.
2
J. Bagaria, N. Castells and P. Larson
the technical background of Ω-logic, and prove some of the basic theorems in this area. This paper assumes a basic knowledge of Set Theory, including constructibility and forcing. All undefined notions can be found in [4].
1. Ω 1.1. Preliminaries Given a complete Boolean algebra B in V , we can define the Boolean-valued model V B by recursion on the class of ordinals On: V0B = ∅ VβB , if λ is a limit ordinal VλB = β
