RIMS Set Theory Workshop 2012
Forcing extensions and large cardinals
December 4 - 7, 2012
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.
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.
- David Aspero (Vienna University of Technology, Austria)
Boban Velickovic (University of Paris 7, France) Cancelled
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: