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

TL;DR

This paper investigates ultrafilters, especially Q-points and selective ultrafilters, within the algebraic structure of the cech-Stone compactification, revealing conditions under which idempotent ultrafilters do not exist, with implications for choiceless set theory.

Contribution

It establishes that certain ultrafilters do not contain idempotent elements and shows that the existence of nonprincipal ultrafilters does not imply idempotent ultrafilters in ZF, answering open questions.

Findings

Q-points and selective ultrafilters lack idempotent elements in their generated families.

In models of ZF without choice, idempotent ultrafilters may not exist.

The existence of nonprincipal ultrafilters does not imply idempotent ultrafilters.

Abstract

We study ultrafilters from the perspective of the algebra in the \v{C}ech-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--Frol\'{\i}k 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 $\omega$ 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.