A note on uniform ultrafilters in a choiceless context
Toshimichi Usuba
TL;DR
This work investigates the behavior of uniform ultrafilters in choiceless set theory, addressing Hayut–Karagila’s questions about how $\mathcal{U}$ can behave at singulars and successors. Using carefully designed symmetric extensions built on large-cardinal assumptions (including supercompact and strongly compact starting points), the authors realize three principal configurations: (i) $\aleph_{\omega+1}$ as the least element of $\mathcal{U}$, (ii) $\aleph_{\omega}$ as the least cardinal not in $\mathcal{U}$, and (iii) $\aleph_{\omega}$ not in $\mathcal{U}$ with every singular $\lambda>\aleph_{\omega}$ of cofinality $\omega$ carrying a uniform ultrafilter. They further show that any attempt to realize mixed patterns (e.g., a gap with $\kappa\notin \mathcal{U}$ but $\lambda\in \mathcal{U}$) entails inner models of measurable cardinals, signaling robust large-cardinal strength. The results illuminate the extent to which $\mathcal{U}$ can be controlled without Choice, and they delineate the boundaries between possibility and necessity in choiceless ultrafilter configurations.
Abstract
In \cite{HK}, Hayut and Karagila asked some questions about uniform ultrafilters in a choiceless context. We provide several answers to their questions.