Ultrafilters over Successor Cardinals and the Tukey Order

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.