Towers in filters, cardinal invariants, and Luzin type families

TL;DR

This paper explores the existence and properties of towers in filters on natural numbers, examining their relation to cardinal invariants, Luzin families, and the independence results within ZFC set theory.

Contribution

It provides new results on which filters contain towers, their consistency with various set-theoretic assumptions, and the relationships between towers, cardinal invariants, and Luzin families.

Findings

Many classical tall filters contain no towers in ZFC.

Tall analytic P-filters can contain towers of arbitrary regular height under consistency assumptions.

All towers can generate non-meager filters, and ultrafilter tower existence is independent of ZFC.

Abstract

We investigate which filters on $\omega$ can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in $[\omega]^\omega$). We prove the following results: - Many classical examples of nice tall filters contain no towers (in ZFC). - It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). - It is consistent that all towers generate non-meager filters, in particular (consistently) Borel filters do not contain towers. - The statement "Every ultrafilter contains towers." is independent of ZFC. Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters ($\mbox{add}^*(\mathcal F)$, $\mbox{cof}^*(\mathcal F)$, $\mbox{non}^*(\mathcal F)$, and $\mbox{cov}^*(\mathcal F)$), and the existence of Luzin type families (of size $\geq \omega_2$), that is, if $\mathcal F$ is a filter then $X\subseteq [\omega]^\omega$ is an $\mathcal F$-Luzin family if $\{A\in X:|A\setminus F|=\omega\}$ is countable for every $F\in \mathcal F$.