# Category Archives: RIMS

## Infinite Combinatorics and Forcing Theory, Kyoto, November 28 – December 1, 2016

RIMS Workshop on Infinite Combinatorics and Forcing Theory
November 28 – December 1, 2016
at the Research Institute for Mathematical Sciences (RIMS), Kyoto University

The workshop website:
http://www.ipc.shizuoka.ac.jp/~styorio/rims16/index.html

Tutorial speakers:

Speakers:

RIMS workshop is held in almost every year at the RIMS, Kyoto University in Kyoto city.
The topic of the workshop is set theory and its applications.
Its goal is to bring together researchers in these areas from Japan
and abroad and to foster academic exchange.
The program will feature two tutorials.
Additionally, we expect many talks, in particular by junior
participants, both from Japan and abroad.

Participants are encouraged to contribute with a talk.
If you are interested, please send an email to the organizer:
Teruyuki Yorioka (Shizuoka University)
styorio@ipc.shizuoka.ac.jp

Organizer:
Teruyuki Yorioka (Shizuoka University)

Scientific Committee:
Joerg Brendle (Kobe University)
Diego A. Mejía (Shizuoka University)

## Recent Developments in Axiomatic Set Theory

September 16 (Wed.)
14:20-14:50 Masaru Kada and Takuto Kato: Variants of AC under ZF minus union
15:10-15:40 Masaru Kada and Souji Shizuma: Some remarks on in nite hat guessing games
16:00-16:50 Makoto Takahashi: On non -shortness of Axiom A posets with frame systems
September 17  (Thu.)
10:00-10:50 Teruyuki Yorioka: Some consistency results with the existence of a non special Aronszajn tree
11:10-12:00 David Chodounsky: F-Mathias reals and generic filters
14:00-14:50 Joel David Hamkins: Upward closure in the generic multiverse of a countable model of set theory
15:10-16:00 Toshimichi Usuba: Set-theoretic geology and large large cardinals
16:20-17:00 Tadatoshi Miyamoto: Side condition methods and morasses

September 18 (Fri.)
9:00-9:50 Sakae Fuchino: On the superuniverse of the set theoretic multiverses
10:10-11:00 Hiroshi Sakai: Covering and approximation properties of ultrapower

Venue

Room 420, RIMS Kyoto

Organizer

Masahiro Shioya (University of Tsukuba), E-mail: shioya _at_ math.tsukuba.ac.jp

# RIMS Set Theory Workshop 2014

Infinitary Combinatorics in Set Theory and Its Applications

November 10 – 13, 2014

Kyoto, Japan

### Overview

Infinitary Combinatorics is a classical and main topic of axiomatic set theory. To investigate infinitary combinatorics, set theory have used and developed not only purely combinatorial arguments but also various methoeds such as forcing, inner model theory, definability and computabity of the structure, and so on. And recently, applications of combinatorics in set theory to other fields of matehmatics are going on increasing. In this RIMS workshop, we shall reexamine recent developments in set theory in light of combinatorics.

As is usually the case in this series of annual RIMS workshops in set theory, besides this main theme of the meeting, there will be also talks on topics in set theory which might not be directly connected to this main theme, as well as talks on some application of set theory in other fields of mathematics.The meeting will take place in the week before Workshop on the Occasion of Sakae Fuchino’s 60th Birthday (Nov. 17 – Nov. 19, 2014; Kobe University, Japan; organizers: Joerg Brendle, Hiroshi Sakai, Toshimichi Usuba), and we expect that many participants from abroad will stay for both meetings.

#### Guest Speakers

• Benedikt Löwe
• Andrew Brooke-Taylor

We expect many talks, in particular by junior participants, both from Japan and abroad. Prospective participants should contact the organizer, Toshimichi Usuba, 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).

## Reflection principles and set theory of large cardinals, September 9 – 12, 2013

The RIMS meeting “Reflection principles and set theory of large cardinals”
will take place from Sept. 9 till Sept. 12 at the Research Institute for
Mathematical Sciences, Kyoto University, Japan. this meeting is organized
by Sakae Fuchino.

#### Guest Speakers

• James Cummings     Mini Course on “Large Cardinals and Forcing”
• Stevo Todorčević     Combinatorial Reflection Principles

We expect many talks, in particular by junior participants, both from Japan and abroad. people interested in attending should contact Sakae Fuchino. fuchino@diamond.kobe-u.ac.jp

12:10–14:00     Lunch Time

 09:00–09:10 Opening 09:10–10:10 James CummingsLarge cardinals and forcing (mini course (1/4)) 10:30–11:15 David Asperó Forcing locally definable well-orders of the universe without the GCH [slides] 11:25–12:10 Laura Fontanella Large properties at small cardinals 14:00–14:45 Michael HrušákGeneric existence of MAD families 15:10–15:55 Teruyuki Yorioka Side condition method and the preservation of a tower in ${\cal P}(\omega)$ 16:10–16:55 Aleksander Błaszczyk $P_\lambda$-filters and regular embeddings

September 10 (Tu)

 09:00–10:00 James CummingsLarge cardinals and forcing (mini course (2/4)) 10:30–11:15 Andrew Brooke-Taylor On cardinal characteristics of large cardinals 11:25–12:10 Diego Alejandro Mejía Guzmán Template iterations with non-definable posets 12:10–13:50 Lunch Time 13:50–14:20 Osvaldo Guzman Namba forcing and set theory of the reals 14:30–15:00 Arturo Antonio Martinez Canjar Ideals 15:20–15:50 Jonathan Cancino Manriquez Countable irresolvable spaces 16:00–16:20 Shohei Tajiri A remark on “Can you take Solovay’s inaccessible away?” 16:30–17:00 David Chodounský Some combinatorics in ${\cal P}(\omega)$ [slides]

September 11 (We)
12:00–13:40     Lunch Time
Short talks session:

 09:00–10:00 James CummingsLarge cardinals and forcing (mini course (3/4)) 10:30–12:00 Stevo Todorčević;Combinatorial Reflection Principles 13:40–14:00 Vincenzo Dimonte Very Large Cardinals and the failure of GCH 14:05–14:25 Luz María García-Ávila A forcing notion related to Hindman’s Theorem (new properties) 14:30–14:50 Sakaé Fuchino A proof of the Fodor-type Reflection Principle from the Rado Conjecture [slides] 14:55–15:15 Toshimichi Usuba Superdestructibility of superstrong and other large cardinals [preprint] 15:20–15:40 Hiroshi Sakai $n$-stationary sets and $\Pi^1_{n-1}$-indescribable cardinals

September 12 (Th)

 09:00–10:00 James CummingsLarge cardinals and forcing (mini course (4/4)) 10:20–11:05 Masahiro Shioya A simplified approach to saturated ideals 11:15–12:00 Tadatoshi Miyamoto Matrices of isomorphic models and morass-like structures 12:10–13:40 Lunch Time 13:40–14:25 Ulises Ariet Ramos García Intersection numbers of families of ideals 14:35–15:20 Liuzhen Wu $\Delta_1$ definability of the full nonstationary ideal on cardinals 15:20–15:30 Closing

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

### 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).