Abstracts, Slides, Preprints, etc.
David Asperó, TU Wien
Wishful thinking in set theory: Omega-completeness and maximality
[slides]
Tristan Bice, Kobe Univ.
Set Theory of the Calkin Algebra
Jörg Brendle, Kobe Univ.
Aspects of splitting
[notes taken by Martin Goldstern]
Andrew Brooke-Taylor, Kobe Univ.
Large cardinals and colimits
[slides]
This is joint work with Joan Bagaria. Colimits take a central role in
category theory, giving natural definitions for many familiar
constructions. If you have a category and a subcategory of interest, a key
question thus is whether colimits are absolute between them. Assuming
suitable large cardinals exist, one can show that for many important cases
they are indeed absolute. Having gone through the definitions starting
from the very start, I will talk about this in the context of the category
of all structures for some language and its subcategory of models for some
theory, and then finish by mentioning more general results.
Vincenzo Dimonte, KGRC
Forcing and Very Large Cardinals: DOs and DON'Ts
[printer version of the slides]
At the top of the large cardinals hierarchy sit the rank-into-rank axioms,
a sequence of hypotheses so strong that they still have no equiconsistent
propositions in other parts of mathematics. Their definition involves the
existence of an elementary embedding from a set (or, in a stronger version,
a class) to itself. The strongest of the rank-to-rank axioms is I0, the
existence of an elementary embedding from $L(V_{\lambda+1})$ to itself with
critical point less than $\lambda$. I0 entails many consequences of the
structure of $L(V_{\lambda+1})$ that, surprisingly, are very similar to
some properties of $L(\mathbb{R})$ under the Axiom of Determinacy. The
application of forcing in this setting, however, has encountered some
resistance. The seminar will be a survey on the applications of forcing at
rank-into-rank axioms, when successful, and on the problems that couldn't
be approached with forcing and that required another strategy.
Vera Fischer, KGRC
Projective wellorders and maximal families of orthogonal measures
with large continuum
[slides]
We study maximal orthogonal families of Borel probability
measures on $2^\omega$ (abbreviated m.o. families) and show that there are
generic extensions of the constructible universe $L$ in which each of the
following holds:
1. There is a $\Delta^1_3$-definable well order of the reals, there is a
$\Pi^1_2$-definable m.o. family, there are no $\bf{\Sigma^1_2}$-definable
m.o. families and $\mathfrak{b}=\mathfrak{c}=\omega_3$ (in fact any
reasinable value of $\mathfrak{c}$ will do).
2. There is a $\Delta^1_3$ definable well order of the reals, there is a
$\Pi^1_2$-definable m.o. family, there are no $\bf{\Sigma^1_2}$-definable
m.o. families, $\mathfrak{b}=\omega_1$ and $\mathfrak{c}=\omega_2$.
Sy-David Friedman, KGRC
The Stable Core of $V$
The stability predicate $S$ is the class of all
triples $(n,\alpha,\beta)$ so that $V_\alpha$ is $\Sigma_n$ elementary
in $V_\beta$. We show that $V$ is generic over $L[S]$ via a forcing
definable over $(L[S],S)$. A corollary is that $V$ is generic
over $HOD$, improving a theorem of Vopenka. The model $L[S]$ is
called the stable core and is more absolute than $HOD$. Many
open questions about it remain.
Sakaé Fuchino, Kobe Univ.
On reflection of list chromatic number
[printer
verson of slides]
[related
preprints and reprints]
This is joint work with Hiroshi Sakai.
The following reflection theorem
is known to be equivalent (over ZFC) to Fodor-type Reflection Principle:
an uncountable graph has countable coloring number if and only
if all of its susgraphs of size $\leq\aleph_{1}$ have countable coloring number.
In particular, this reflection theorem is independent from ZFC+CH or ZFC+$\neg$CH.
On the other hand, we have the following non reflection theorem (in ZFC)
for countable chromatic number of graphs: for any $1\leq n\lt\omega$ there
is a graph $G$ of uncountable chromatic number such that all
subgraphs of cardinality
$\lt\beth_{n}$ are of countable chromatic number. (Erdös & Hajnal 1961).
In this talk we show that the reflection and non-reflection of
countable list-chromatic number of graphs follow a pattern
different form both of the theorems above.
Hiroshi Fujita, Ehime Univ.
Borel-generated subgroups of the real line
Martin Goldstern, TU Wien
Ultralaver forcing
[notes for the talk]
Ultralaver forcing is a ccc variant of Laver forcing. Conditions are trees
in which the successor sets of all nodes $s$ above the stem are not only
infinite but lie in some ultrafilter $U_s$. I will discuss some
basic properties of this forcing and sketch how it can be used
in a ccc iteration to obtain a model for the Borel Conjecture.
Masaru Kada, Osaka Prefecture Univ.
Remarks on Scheepers' theorem on the cardinality of Lindelöf
spaces
[slides] (revised version on Tue,
31 Jan 2012 09:52:00 +0900)
Scheepers proved the following theorem:
If the existence of a measurable cardinal is consistent,
then it is consistent that every points-$G_\delta$
indestructibly Lindelöf space has cardinality at most $2^{\aleph_0}$.
In this paper, we will review the proof of Scheepers' theorem
and remark that we can slightly improve the statement of the main theorem
in Scheepers' paper and yet significantly simplify its proof.
Jakob Kellner, KGRC
Borel Conjecture and dual Borel Conjecture
I give a superficial overview of the proof in
Borel Conjecture and
Dual Borel Conjecture
http://arxiv.org/abs/1105.0823
(joint with Goldstern, Shelah and Wohofsky), roughly following http://arxiv.org/abs/1112.4424.
Essential ingredients of the construction are ultralaver and Janus forcings,
which will also be discussed in the talks of Goldstern and Wohofsky, respectively.
Taishi Kurahashi, Kobe Univ.
Predicate modal logic of provability and Montagna's problem
For any recursively enumerable theory $T$, there exists a $\Sigma_1$
formula ${\rm Pr}_T(x)$ which represents the relation ``$x$ is provable in
$T$'' in $T$. A mapping $*$ from the set of all predicate modal formulas to
the set of all formulas in the language of $T$ is called $T$-interpretation
if $*$ preserves all Boolean connectives and $*$ maps $\Box A$ to ${\rm
Pr}_T(\ulcorner A^* \urcorner)$. Let ${\sf PL}(T)$ be the set of all modal
formulas provable in $T$ for any $T$-interpretation. Montagna conjectured
that the system of predicate modal logic ${\sf QGL}$ characterizes the set
$\bigcap \{{\sf PL}(T)|T$ is a recursively enumerable extension of ${\sf
PA}\}$. I introduce some topics around Montagna's problem.
Diego Alejandro Mejía, Kobe Univ.
Matrix iterations and the Cichon's diagram
[slides] [notes]
The matrix iteration technique was presented for the first
time by A. Blass and S. Shelah in 1985 to prove the Independence between
the dominating number and the ultrafilter number. Later, in 2011, J.
Brendle and V. Fischer used this technique to prove consistency results
about the relation between the unbounding number, the splitting number and
the almost-disjoint number.
In this talk, using this technique, I present consistency results about the
cardinals on the right hand side of the Cichon's diagram, proving that they
may take arbitrary values when we use at most three regular cardinals.
This is part of my doctoral research under the supervision of professor J.
Brendle.
Hiroaki Minami, KGRC
Reaping number and independence number for partitions of $\omega$.
[slides]
The structure $([\omega]^{\omega},\subset^{*}) $ is well studied in
set theory. The combinatorial property of
$([\omega]^{\omega},\subset^{*}) $ is described by cardinal invariants
and their relationship has been studied.
In recent years partial orders similar to
$([\omega]^{\omega},\subset^{*}) $ have been focused on and analogue
cardinal invariants have been defined and investigated.
In this talk, we study the relation between reaping number and
independence number for the structure $((\omega)^{\omega},\leq^{*})$,
the set of all infinite partitions of $\omega$ ordered by "almost
coarser". By using iterated forcing along templates, we will show the
reaping number is smaller than the independence number for partitions
of $\omega$.
Miguel Angel Mota, KGRC
Generalizations of Martin's Axiom and their
consistency with a large continuum.
[slides]
We prove that the restriction of PFA to the class of the small
proper posets which are finitely proper is consistent with a large
continuum. We also introduce a new generalization of the full Martin's
Axiom and sketch the prove of its consistency. This is joint work with D.
Asperó.
Dilip Raghavan, Kobe Univ.
Suslin lattices
A Suslin lower semi lattice is an uncountable well founded lower
semi-lattice with no uncountable chains or antichains. Every normal Suslin
tree is a Suslin lower semi-lattice. But there can be Suslin lower
semi-lattices that are quite different from Suslin trees. We will present
what is known about these objects as well as what remains open. This is
joint work with Teruyuki Yorioka.
Hiroshi Sakai, Kobe Univ.
$\omega_1$ stationary preserving inproper poset of size $\omega_1$
Chang's Conjecture and weak square
[slides]
We discuss how weak square principles are
denied by Chang's Conjecture and its generalizations.
Among other things we prove that Chang's Conjecture
is consistent with $\square_{\omega_1 , 2}$.
Frank Tall, Univ. of Toronto
P. Burton, R. Dias, F. Tall*
Lindelof Indestructibility, Topological games, and Selection Principles: New Results
[slides]
I and my students have new results concerning topological games
and selection principles related to the question of when
Lindelofness is preserved by countably closed forcing.
Victor Torres, KGRC
Conjectures of Rado and Chang, and Special Aronszajn Trees
[slides]
We show that both Rado's Conjecture and strong Chang's Conjecture
imply that there are no special $\aleph_2$-Aronszajn trees if
the Continuum Hypothesis fails. We give similar
result for trees of higher heights and
we also investigate the influence of Rado's Conjecture on square sequences
Toshimichi Usuba, Nagoya Univ.
Small semiproper posets
[slides]
For an infinite cardinal $\kappa$,
let $(\dagger)_\kappa$ be the assertion that
every $\omega_1$-stationary preserving poset with size equal or less than $\kappa$
is semiproper.
We show that $(\dagger)_{\omega_2}$ is a strong property which
implies a kind of strong Chang's Conjecture.
We also study other consequences from $(\dagger)_{\kappa}$.
Andrés Villaveces, UN Colombia
Forcing and Categoricity
In recent years the interaction between Set Theory and Model
Theory has increased due to renewed attention to model theoretic contexts
outside of First Order Logic. Whereas good parts of First Order Logic tend
to be absolute, important questions in infinitary logics, abstract
elementary classes, etc. (categoricity, universality among others) depend
up to some point on the model of set theory where one is working. Forcing
techniques play a central role in drawing the line between absolute and
relative model theoretic notions. I will present recent results (some of
them joint work with Asperó) on the interaction between Forcing and
Categoricity in non-elementary variants of Random Bipartite Graphs
(axiomatized in the infinitary logic with generalized quantifiers
$L_{\omega_1,\omega}(Q)$). I will also focus on dichotomies between
combinatoric principles (weak diamonds) and forcing axioms/forcing
constructions.
Wolfgang Wohofsky, TU Wien
Janus forcing towards dual Borel Conjecture, and other variants of the Borel Conjecture
Janus forcings are a family of forcing notions designed to obtain the
dual Borel Conjecture (in our model of Borel Conjecture and
dual Borel Conjecture). Janus forcings have two faces (therefore the
name): On the one hand, they are "sufficiently close to" Cohen
forcing, which allows to carry out a proof which is reminiscent
of Carlson's original proof of the dual Borel Conjecture. On the other
hand, they "can be made into" random forcing which is central for
not destroying the Borel Conjecture.
If there is time left (probably not much), I will also briefly
discuss another variant of the Borel Conjecture (Marczewski
Borel Conjecture), and its connection to the notion of Sacks dense ideal
(which I introduced to study whether it is consistent).
Liu Zhen Wu, KGRC
$\Delta_1$-definability of the nonstationary ideal
Teruyuki Yorioka, Shizuoka Univ.
Killing some $S$-spaces by a coherent Suslin tree
[slides]
[preprint]
PFA(S), which is introduced by Stevo Todorcevic, is an axiom that
there exists a coherent Suslin tree S such that
the forcing axiom holds for every proper forcing
which preserves S to be Suslin.
It is an open question whether under PFA(S),
S forces that there are no S-space.
Yasuo Yoshinobu, Nagoya Univ.
On weakly operationally closed posets
[slides]
We introduce a property of posets named `weakly operationally closed', defined
in terms of a modified
version of generalized Banach-Mazur games, and show that PFA is
preserved under forcing over posets with this property.
Lyubomyr Zdomskyy, KGRC
Productively Lindelöf spaces and selection principles
Shuguo Zhang, Sichuan Univ.
Cardinal invariants related to $I$-ultrafilters
Last modified: Thu Mar 01 23:39:38 +0900 2012