Archives of: Kurt Godel Research Center

David Asperó – Special $\aleph_2$-Aronszajn trees and GCH

Talk held by David Asperó (University of East Anglia, Norwich, UK) at the KGRC seminar on 2018-10-22.

Abstract: In joint work with Mohammad Golshani, and assuming the existence of a weakly compact cardinal, we build a forcing extension in which GCH holds and every $\aleph_2$-Aronszajn tree is special. This answers a well-known question from the 1970’s. I will give the proof of this theorem, with as many details as possible.

Monroe Eskew – Rigid ideals

Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-10-18.

Abstract: Using ideas from Foreman-Magidor-Shelah, one can force from a Woodin cardinal to show it is consistent that the nonstationary ideal on $\omega_1$ is saturated while the quotient boolean algebra is rigid. The key is to apply Martin’s Axiom to the almost-disjoint coding forcing to see how it interacts with a generic elementary embedding. This strategy requires the continuum hypothesis to fail. Towards showing the consistency of rigid ideals with GCH, the speaker investigated other coding strategies: stationary coding (with Brent Cody), a rigid version of the Levy collapse, and ladder-system coding (in recent work with Paul Larson). We have some equiconsistencies about rigid ideals on $\omega_1$ and $\omega_2$, as well as some global possibilities from very large cardinals. Some natural questions remain about $\omega_1$  and successors of singulars.

David Chodounský – Silver forcing and P-points

Research seminar, Kurt Gödel Research Center – Thursday, October 11th

Abstract: I will give a full proof of a joint result with O. Guzman regarding a
technique for destroying P-ultrafilters with Silver forcing. Time
permitting, I will present several applications.

Gerhard Jäger – From fixed points in weak set theories to some open problems

Research seminar, Kurt Gödel Research Center –  October 9th

Abstract: Least fixed points of monotone operators are well-studied objects in many
areas of mathematical logic. Typically, they are characterized as the
intersection of all sets closed under the respective operator or as the
result of its iteration from below.

In this talk I will start off from specific $\Sigma_1$ operators in a
Kripke-Platek environment and relate fixed point assertions to alternative
set existence principles. By doing that, we are also led to some
“largeness axioms” and to several open problems.

David Fernández-Bretón – Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Talk held by David Fernández-Bretón (KGRC) at the KGRC seminar on 2018-10-04. The recorded talk is available here.

Abstract: In the absence of the Axiom of Choice, there may be infinite sets for which certain Ramsey-theoretic statements – such as Ramsey’s or (appropriately phrased) Hindman’s theorem – fail. In this talk, we will analyse the existence of such sets, and their precise location within the hierarchy of infinite Dedekind-finite sets; independence proofs will be carried out using the Fränkel-Mostowski technique of permutation models.

This is joint work with Joshua Brot and Mengyang Cao.

Zoltán Vidnyánszky: Borel chromatic numbers: basis and antibasis results

Talk held by Zoltán Vidnyánszky (KGRC) at the KGRC seminar on 2018-06-14.

Abstract: We give a full description of the existence of a homomorphism basis for Borel graphs of given Borel chromatic number. In particular, we show that there is a Borel graph with Borel chromatic number 3 that admits a homomorphism to any Borel graph of Borel chromatic number at least 3. We also discuss the relation of these results to Hedetniemi’s conjecture.

Andrea Medini: Homogeneous spaces and Wadge theory

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2018-06-07.

Abstract: All spaces are assumed to be separable and metrizable. A space $X$ is homogeneous if for all $x,y\in X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$. A space $X$ is strongly homogeneous if all non-empty clopen subspaces of $X$ are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of $2^\omega$.

This is joint work with Raphaël Carroy and Sandra Müller.

Borisha Kuzeljevic: P-ideal dichotomy and some versions of the Souslin Hypothesis

Talk held by Borisha Kuzeljevic (Czech Academy of Sciences, Prague) at the KGRC seminar on 2018-05-24.

Abstract: The talk will be about the relationship of PID with the statement that all Aronszajn trees are special. This is joint work with Stevo Todorcevic.

Yair Hayut: Stationary reflection at $\aleph_{\omega+1}$

Talk held by Yair Hayut (Tel-Aviv University, Israel)
at the KGRC seminar on 2018-05-17.

Abstract: Stationary reflection is one of the basic prototypes of reflection phenomena,
and its failure is related to many counterexamples for compactness
properties (such as almost free non-free abelian groups,
and more). In 1982, Magidor showed that it is consistent, relative to infinitely many
supercomapct cardinals, that stationary reflections holds at $\aleph_{\omega + 1}$.
In this talk I’m going to present a new method for forcing stationary reflection
at $\aleph_{\omega+1}$, which allows to significantly reduce the upper bound for the consistency strength of the full stationary reflection at $\aleph_{\omega+1}$ (below a single partially supercompact cardinal).

This is a joint work with Spencer Unger.

Vincenzo Dimonte: The sensitive issue of iterability

Talk held by Vincenzo Dimonte (University of Udine, Italy) at the KGRC seminar on 2018-05-03.

Abstract: In the momentous years when the community of set theorists was reaching the definite answer for the problem of the consistency of the Axiom of Determinacy, Martin wrote a small paper in the Proceedings of the International Congress of Mathematicians, 1978, in which he proved that the iterable version of I3, a very large cardinal, implied the determinacy of $\Pi^1_2$ sets of reals. Later it was proved that AD had much lower consistency, and iterable I3 fell into oblivion. In the last decade interest on I3 re-emerged, but iterable I3 is still elusive, and the small paper by Martin is not helpful, as it is terse and full of gaps. Even the definition of iterable I3 is not convincing. In this seminar we will bring back to life this abandoned hypothesis, clean it up to modern standards, and reveal the existence of a new hierarchy of axioms that was previously overlooked.