JournalsowrVol. 5 / No. 1DOI 10.4171/owr/2008/02

Set Theory

  • Sy-David Friedman

    Universität Wien, Austria
  • Menachem Magidor

    The Hebrew University, Jerusalem, Israel
  • W. Hugh Woodin

    University of California, Berkeley, United States
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Abstract

This was a highly successful workshop on some of the exciting recent developments in set theory. There were 45 participants. By limiting the number of talks to 25, there was ample time for research collaboration in small groups, which rendered the meeting very relaxed, enjoyable and productive.

Among the highlights of the workshop are the following:

Justin Moore presented some dramatic new work on the existence of a universal Aronszajn line, demonstrating that the existence of a canonical, universal linear ordering is not limited to the countable case. There were several beautiful presentations in the area of descriptive set theory: Christian Rosendal provided a rough classification of Banach spaces, Slawomir Solecki described G-delta ideals of compact sets, Simon Thomas drew some striking connections between set theory and geometric group theory and John Clemens discussed Borel homomorphisms of smooth Sigma-ideals. A major new result was presented by David Schrittesser, who showed that projective measurability does not imply the projective Baire property.

There were two major results in the related areas of large cardinals and determinacy: John Steel calculated the consistency strength of determinacy for games on reals and he together with Ronald Jensen discovered a way of building the core model for a Woodin cardinal in ZFC without additional large cardinal hypotheses. Several talks presented important recent work related to the pcf theory: Todd Eisworth discussed combinatorial properties at the successor of a singular cardinal, Matteo Viale told us about reflection and approachability and Moti Gitik brought us up to date on consistency results regarding the behaviour of the generalised continuum function.

In combinatorial set theory we heard lectures on induced Ramsey theory, MAD families, weak diamonds and simple cardinal invariants. And in the theory of forcing we heard about Laver products, p-points, indestructibility of weak compactness, mixed support iterations and reflection.

Paul Larson's very nice talk on Martin's Maximum and definability rounded out a very successful meeting.