Kurt Gödel Research Center

2015 seminar talk: Towers in Borel filters

Talk held by Barnabás Farkas (KGRC) at the KGRC seminar on 2015-11-19.

Abstract

In a joint work with J. Brendle and J. Verner we studied which ultrafilters and which Borel filters can contain a tower, that is, a $\subseteq^*$-decreasing sequence in the filter without a pseudointersection in $[\omega]^\omega$.

First, I will give a short survey on the following result: The statement “every ultrafilter contains a tower” is independent from ZFC.

Then my talk will be focused mainly on Borel filters and on some selected results concerning possible logical implications between (i) the existence of towers in certain classical Borel filters, (ii) inequalities between cardinal invariants of these filters, and (iii) the existence of a peculiar object, a large $\mathcal{F}$-Luzin set, that is, a family $\mathcal{X}\subseteq [\omega]^\omega$ of cardinality $\geq\omega_2$ such that $\{X\in\mathcal{X}: X\nsubseteq^* F\}$ is countable(!) for every $F\in\mathcal{F}$ (where $\mathcal{F}$ is a filter).