Forcing extensions and large cardinals, December 4 – 7, 2012

Dates, venue, and organizer

Overview

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 axiomto 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.

Minicourse

  • David Aspero   (Barcelona Set Theory Group, Spain)
    “Iterated forcing with side conditions”

 

Other talks

  • Teruyuki Yorioka   (Shizuoka University)
    “The omega properness, club guessing and PFA(S)”
  • Teruyuki Yorioka   (Shizuoka University)
    “A comment on Aspero-Mota iteration”
  • Toshimichi Usuba   (Nagoya University)
    “Partial stationary reflection principles”
  • Yasuo Yoshinobu   (Nagoya University)
    “Operations vs. *-tactics”
  • Diego Mejia   (Kobe University)
    “Models of some cardinal invariants with large continuum”
  • Hiroaki Minami   (Nagoya University)
    “Reaping number and independence number for partitions of ω”
  • Daisuke Ikegami   (University of California, USA)
    “Ω-logic and Boolean valued second order logic”
  • Masaru Kada   (Osaka Prefecture University)
    “A technique for proving preservation of topological properties under forcing extensions”
  • Andrew Brooke-Taylor   (Kobe University)
    “Weak squares and subcompact cardinals”
  • Tadatoshi Miyamoto   (Nanzan University)
    “Proper forcing with side conditions”
  • Jörg Brendle   (Kobe University)
    “Almost disjoint families built from closed sets”

 

This workshop is part of a series of workshops held in Japan every year and supported by the Research Institute for Mathematical Sciences (RIMS).

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