RIMS Set Theory Workshop 2012

Forcing extensions and large cardinals

December 4 - 7, 2012

Kyoto, Japan

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By strengthening the standard Zermelo-Fraenkel axiom system of set theory (ZFC), one can decide a number of important statements in mathematics, like e.g. the continuum problem. For example, adding a certain kind of forcing axiom to ZFC makes the continuum have size aleph_2, the second uncountable cardinal. A classical result of S. Todorcevic and B. Velickovic says that the proper forcing axiom PFA is an axiom of this kind. To see whether such a forcing axiom may be added to ZFC, i.e., whether it is consistent with ZFC, one needs to build a model in which the additional axiom holds. For doing this, one starts with a ground model of ZFC with large cardinals and extends this model by using iterated forcing. That is, by repetitively adjoining the needed objects in a generic fashion one finally obtains a model satisfying the forcing axiom. However, this method does not always work well. That it does in case of proper and semiproper forcing has been shown by S. Shelah.

An interesting and important problem asks what can happen to the size of the continuum if a strong forcing axiom like PFA is weakened. To answer this kind of problem we need to control the behavior of reals in iterated forcing extensions. Apart from a few special cases, this is a difficult problem, one difficulty being to understand what happens in limit stages of iterated forcing. Recently, David Aspero and Miguel Mota have developed a new approach to iterated forcing in which the generation of new reals is controled by side conditions incorporated directly into the iteration.

The topic of this meeting are such new approaches to iterated forcing. Its goal is to bring together researchers from Japan and abroad and to foster academic exchange. The program will feature talks by the participants and discussion sessions.

Guest Speakers

We expect many talks, in particular by junior participants, both from Japan and abroad. Prospective participants should contact the organizer, Tadatoshi Miyamoto, as early as possible.

Dates, venue, and organizer

This workshop is part of a series of workshops held in Japan every year and supported by the Research Institute for Mathematical Sciences (RIMS). Here are web pages of the past four years:
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