Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory

Abstract

We study ultrafilters from the perspective of the algebra in the Čech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a selective ultrafilter) and ${\mathcal{F}}^p$ (resp. ${\mathcal{G}}^p$) is the smallest family containing $p$ and closed under iterated sums (resp. closed under Blass--Frol\'ık sums and Rudin--Keisler images), then ${\mathcal{F}}^p$ (resp. ${\mathcal{G}}^p$) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form $\mathbf{L}(\mathbb{R})[p]$ where $\mathbf{L}(\mathbb{R})$ is a Solovay model (of $\mathsf{ZF}$ without choice) and $p$ is a selective ultrafilter, there are no idempotent elements. In particular, the theory $\mathsf{ZF}$ plus the existence of a nonprincipal ultrafilter on $\omega$ does not imply the existence of idempotent ultrafilters, which answers a question of DiNasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397-411). Following the line of obtaining independence results in $\mathsf{ZF}$, we finish the paper by proving that $\mathsf{ZF}$ plus "every additive filter can be extended to an idempotent ultrafilter" does not imply the Ultrafilter Theorem over $\mathbb{R}$, answering another question of DiNasso and Tachtsis from the same paper.

27 pages, a few minor typos corrected from the previous version