Ultrafilters over Successor Cardinals and the Tukey Order

TL;DR

This paper investigates ultrafilters on uncountable regular cardinals, especially , analyzing their Tukey order relations, independence results, and constructions under various set-theoretic assumptions.

Contribution

It provides new constructions of ultrafilters extending the club filter, analyzes Tukey equivalences under PFA, and explores independence results related to ultrafilters over .

Findings

Ultrafilters over  can be Tukey-equivalent to ^{<\u03a9} under certain conditions.

The Tukey type of Todorcevic's ultrafilter  is characterized under PFA.

Existence of coherent Aronszajn trees compatible with certain ultrafilter extensions.

Abstract

We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform ultrafilter over $\omega_1$ is Tukey-equivalent to $[2^{\aleph_1}]^{<\omega}$, and for each cardinal $\kappa$ of uncountable cofinality, a new construction of a uniform ultrafilter over $\kappa$ which extends the club filter and is Tukey-equivalent to $[2^\kappa]^{<\omega}$. We also analyze Todorcevic's ultrafilter $\mathcal{U}(T)$ under PFA, proving that it is Tukey-equivalent to $[2^{\aleph_1}]^{<\omega}$ and that it is minimal in the Rudin-Keisler order with respect to being a uniform ultrafilter over $\omega_1$. We prove that, unlike PFA, $\text{MA}_{\omega_1}$ is consistent with the existence of a coherent Aronszajn tree $T$ for which $\mathcal{U}(T)$ extends the club filter. A number of other results are obtained concerning the Tukey order on uniform ultrafilters and on uncountable directed systems.