On $\omega$-Strongly Measurable Cardinals

TL;DR

This paper explores the consistency of $\omega$-strongly measurable cardinals within HOD, demonstrating that under certain large cardinal hypotheses, all successors of regular cardinals can be $\omega$-strongly measurable in HOD.

Contribution

It establishes the relative consistency of $\omega$-strongly measurable cardinals in HOD under weaker large cardinal assumptions than previously known.

Findings

Every successor of a regular cardinal can be $\omega$-strongly measurable in HOD.

Consistency results are proved relative to large cardinal hypotheses.

The results weaken the large cardinal assumptions needed for such measurability.

Abstract

We prove several consistency results concerning the notion of $\omega$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa) = \kappa$, that every successor of a regular cardinal is $\omega$-strongly measurable in HOD.