Archives of: Kurt Godel Research Center

Stevo Todorcevic: Ramsey degrees of topological spaces

Talk held by Stevo Todorcevic (University of Toronto, Canada) at the KGRC seminar on 2018-12-04.

Abstract: This will be an overview of structural Ramsey theory when the objects are topological spaces. Open problems and directions for further research in this area will also be examined.

Daniel Soukup: New aspects of ladder system uniformization

Talk held by Daniel Soukup (KGRC) at the KGRC seminar on 2018-11-29.

Abstract: Given a tree $T$ of height $\omega_1$, we say that a ladder system colouring $(f_\alpha)_{\alpha\in \lim\omega_1}$ has a $T$-uniformization if there is a function $\varphi$ defined on a subtree $S$ of $T$ so that for any $s\in S_\alpha$ of limit height and almost all $\xi\in dom (f_\alpha)$, $\varphi(s\upharpoonright \xi)=f_\alpha(\xi)$.

In sharp contrast to the classical theory of uniformizations on $\omega_1$, J. Moore proved that CH is consistent with the statement that any ladder system colouring has a $T$-uniformization (for any Aronszajn tree $T$). Our goal is to present a fine analysis of ladder system uniformization on trees pointing out the analogies and differences between the classical and this new theory. We show that if $S$ is a Suslin tree then CH implies that there is a ladder system colouring without $S$-uniformization, but $MA(S)$ implies that any ladder system colouring has even an $\omega_1$-uniformization.

Furthermore, it is consistent that for any Aronszajn tree $T$ and ladder system $\mathbf C$ there is a colouring of $\mathbf C$ without a $T$-uniformization; however, and quite surprisingly, $\diamondsuit^+$ implies that for any ladder system $\mathbf C$ there is an Aronszajn tree $T$ so that any monochromatic colouring of $\mathbf C$ has a $T$-uniformization. We also prove positive uniformization results in ZFC for some well-studied trees of size continuum. (cf. https://arxiv.org/abs/1806.03867 and https://arxiv.org/abs/1803.03583)

Raphaël Carroy – A dichotomy for topological embeddability between continuous functions

Talk held by Raphaël Carroy (KGRC) at the KGRC seminar on 2018-11-22.

Abstract

We say a function $f$ embeds (topologically) in a function $g$
when there are two topological embeddings $\sigma$ and $\tau$
satisfying $\tau \circ f = g \circ \sigma$. I will prove the
following dichotomy: the quasi-order of topological embeddability
between continuous functions on compact zero-dimensional Polish spaces
is either an analytic complete quasi-order, or a well-quasi-order.

This is a joint work with Yann Pequignot and Zoltán Vidnyánszky.

Russell G. Miller: Hilbert’s Tenth Problem for Subrings of the Rational Numbers

Talk held by Russell Miller (Queens College, City University of New York (CUNY), USA) at the KGRC seminar on 2018-11-15.

 

Abstract: When considering subrings of the field $\mathbb{Q}$ of rational numbers, one can view Hilbert’s Tenth Problem as an operator, mapping each set $W$ of prime numbers to the set $HTP(R_W)$ of polynomials in $\mathbb {Z}[X_1,  X_2, \ldots]$ with solutions in the ring $R_W = \mathbb{Z}[W^{-1}]$. The set $HTP(R_{\emptyset})$ is the original Hilbert’s Tenth Problem, known since 1970 to be undecidable. If $W$ contains all primes, then one gets $HTP(\mathbb{Q})$, whose decidability status is open. In between lie the continuum-many other subrings of $\mathbb{Q}$.

We will begin by discussing topological and measure-theoretic results on the space of all subrings of $\mathbb{Q}$, which is homeomorphic to Cantor space. Then we will present a recent result by Ken Kramer and the speaker, showing that the HTP operator does not preserve Turing reducibility. Indeed, in some cases it reverses it: one can have $V <_T W$, yet $HTP(R_W) <_T HTP(R_V)$. Related techniques reveal that every Turing degree contains a set $W$ which is \emph{HTP-complete}, with $W’ \le_1 HTP(R_W)$. On the other hand, the earlier results imply that very few sets $W$ have this property: the collection of all HTP-complete sets is meager and has measure $0$ in Cantor space.

Thilo Weinert: On Order-Types in Polarised Partition Relations

Talk held by Thilo Weinert (KGRC) at the KGRC seminar on 2018-11-08.

Abstract: The history of the polarised partition relation goes back to the original seminal paper by Erdős and Rado from 1956. For the ordinary partition relation after some time one also investigated order-types. For the polarised partition relation however, I am only aware of three papers where order-type played a role and here nothing beyond well-orders ever seems to have been considered.

We are attempting to rectify this. We will present an analogue of a theorem of Jones and a potentially vacuous generalisation of a proposition of Garti and Shelah. Furthermore we will show limits to further generalisations and analogues and will exhibit some open problems.

This is joint work with Lukas Daniel Klausner.

David Schrittesser: The Ramsey property, MAD families, and their multidimensional relatives

Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-10-25.

Abstract: Suppose every set of real numbers has the Ramsey property and “uniformization on Ellentuck-comeager sets” as well as Dependent Choice hold (as is the case under the Axiom of Determinacy, but also in Solovay’s model). Then there are no MAD families. As it turns out, there are also no (Fin x Fin)-MAD families, where Fin x Fin is the two-dimensional Fubini product of the ideal of finite sets. We also comment on higher dimensional products.

All results are joint work with Asger Törnquist.

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.