Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory
David Fernández-Bretón, Jareb Navarro-Castillo, Jesús A. Soria-Rojas
TL;DR
The paper analyzes idempotent ultrafilters in $\beta\omega$ through Q-points and selective ultrafilters, revealing that iterated sums from a Q-point (and from a selective ultrafilter via Blass–Frolík sums and RK images) fail to produce idempotents in ZFC. It connects these structural results to choiceless set theory, showing that under Blass’s conjecture, models like $\mathbf{L}(\mathbb{R})[p]$ can have nonprincipal ultrafilters but no idempotents, and establishes a ZF consistency result where every additive filter extends to an idempotent ultrafilter even as $\mathrm{UT}(\mathbb{R})$ fails. The work highlights the independence phenomena surrounding idempotent ultrafilters, the behavior of $u$-limits, and the interplay between ultrafilter algebra and choice principles. Overall, it provides both ZFC machinery and ZF/choiceless-model applications that answer questions of Di Nasso and Tachtsis and illuminate the landscape of idempotents without relying on the Ultrafilter Lemma.
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 $\mathscr F^p$ (resp. $\mathscr G^p$) is the smallest family containing $p$ and closed under iterated sums (resp. closed under Blass--Frolík sums and Rudin--Keisler images), then $\mathscr F^p$ (resp. $\mathscr G^p$) 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 $ω$ 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.