David Asperó, TU Wien

[slides]

[notes taken by Martin Goldstern]

[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

[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.

[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 $\mathfrak{c}=\omega_2$.

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.

[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:

On the other hand, we have the following non reflection theorem (in ZFC) for countable chromatic number of graphs:

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.

[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.

[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.

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.

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.

[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.

[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

[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.

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.

[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 $\square_{\omega_1 , 2}$.

P. Burton, R. Dias, F. Tall*

[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.

[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 inï¬‚uence of Rado's Conjecture on square sequences

[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 $(\dagger)_{\kappa}$.

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.

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).

[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.

[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.

Last modified: Thu Mar 01 23:39:38 +0900 2012