RIMS Set Theory Worksop 2013
Reflection principles and set theory
of large cardinals
Abstracts
Ulises Ariet Ramos García
Intersection numbers of families of ideals
We study the intersection number of families of tall ideals.
We show that the intersection number of the
class of analytic $P$-ideals is equal to the bounding number $\mathfrak{b}$,
the intersection number of the class of all meager ideals is equal to
$\mathfrak{h}$ and the intersection number of the class of all $F_{\sigma}$
ideals is between $\mathfrak{h}$ and $\mathfrak{b}$, consistently different
from both.
This is joint work with M. Hrušák, C. A. Martínez-Ranero and O. A.
Téllez-Nieto.
Sakaé Fuchino
A proof of the Fodor-type Reflection Principle from the Rado Conjecture
In this talk I shall sketch a proof of the Fodor-type Reflection
Principle (FRP) from the Rado Conjecture (RC) which is a modification of the proof
of Chang's Conjecture from RC by Todorčević [1].
[1] Stevo Todorčević, Conjectures of Rado and Chang and cardinal
arithmetic, in: N.W. Sauer et al. (eds.), Finite and Infinite
Combinatorics in Sets and Logic. 485-398.
Aleksander Błaszczyk
$P_\lambda$-filters and regular embeddings
We prove that if the topology on the set ${\rm Seq}=\bigcup\{{}^n\omega:n<\omega\}$
of all finite sequences of natural numbers is appointed by $P_\lambda$-filters
and $\lambda<\mathfrak{b}$,
then Seq is a $P_\lambda$-set in its Čech–Stone compactification. This improves a
result of Simon as well as a result of Juhász and Szymański. As a result
we obtained an extension of a result of Burke concerning real functions. We
also partially answer a question posed by him.
David Chodounský
Some combinatorics in ${\cal P}(\omega)$
I will recall definitions of tight, strong and peculiar $(\omega_1,\omega_1)$ gaps
in $\langle \mathcal{P}(\omega), \subset^* \rangle$ and
$\langle {}^\omega \omega, <^* \rangle$. We will discuss relations among
these objects and conditions for their existence.
Diego Alejandro Mejía Guzmán
Template iterations with non-definable posets
As an application of Shelah's template iteration technique
extended to non-definable posets, we prove that, if $\kappa$ is a
measurable cardinal and $\theta<\kappa<\mu<\lambda$ are uncountable regular
cardinals, then there is a ccc poset forcing
$\mathfrak{s}=3D\theta<\mathfrak{b}=3D\mu<\mathfrak{a}=3D\lambda$.
Vincenzo Dimonte
Very Large Cardinals and the failure of GCH
The seminar will describe recent results, in collaboration with Sy
Friedman, about the consistency of several instances of not GCH with very
large cardinals.
Hiroshi Sakai
$n$-stationary sets and $\Pi^1_{n-1}$-indescribable cardinals
Recently the notion of $n$-stationary sets and $n$-$2$-stationary sets
appear frequently in the proof theory (ordinal analysis and topological semantics
of poly-modal logic). In this talk we show that these notions are
equivalent to $\Pi^1_{n-1}$-indescribability in the constructible
universe $L$. We also discuss the consistency strength of the existence
of a cardinal which is $n$-stationary or $n$-$2$-stationary.
This is a joint work with J. Bagaria and M. Magidor.
Stevo Todorčević (Univ. of Toronto and CNRS Paris)
Combinatorial Reflection Principles
This will be a broad overview of the recent work on the
set-theoretic reflection principles in several contexts such as, for
example, stationarity of sets, chromatic numbers of graphs, etc. We shall
also expose some of the open problems in this area.
Toshimichi Usuba
Superdestructibility of superstrong
and other large cardinals
We prove that superstrong cardinals and other large cardinals, such
as extendible cardinals, are not Laver indestructible, that is, these large
cardinals are destructible by directed closed forcing. In fact, these large cardinals are
superdestructible.
Teruyuki Yorioka
Side condition method and the preservation of a tower in ${\cal P}(\omega)$
I mention a brief history of "PID and no $S$-spaces" and
side condition method.
It is proved that some ideal based forcings (which are of form side
condition method,
defined by J. Zapletal) preserves a tower in $\mathcal{P}(\omega)/{\rm fin}$.
Therefore it is also proved that
it is consistent that
PID holds, the pseudo-intersection number is equal to $\aleph_1$ and
there are no $S$-spaces.
This answers a question due to S. Todorcevic.
Last modified: Fri Sep 27 12:34:28 +0900 2013