On the cofinality of the least $\lambda$-strongly compact cardinal

TL;DR

This paper explores the possible cofinalities of the least -strongly compact cardinal, showing consistency results relative to supercompact cardinals and establishing provable properties about its cofinality.

Contribution

It characterizes the cofinalities of the least -strongly compact cardinal, linking them to ultrafilters and large cardinal assumptions.

Findings

Cofinalities can be any regular cardinal with a -complete ultrafilter, consistent with supercompact cardinals.

The cofinality of the least -strongly compact cardinal always carries a -complete uniform ultrafilter.

The paper establishes both consistency and provable properties regarding the cofinality.

Abstract

In this paper, we characterize the possible cofinalities of the least $\lambda$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta$, that carries a $\lambda$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $\delta$, that the least $\lambda$-strongly compact cardinal has cofinality $\delta$. On the other hand, provably the cofinality of the least $\lambda$-strongly compact cardinal always carries a $\lambda$-complete uniform ultrafilter.