On $\omega$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions

TL;DR

This paper investigates the properties of certain large cardinals within $ ext{P}_{ ext{max}}$ extensions of models with determinacy, showing that specific club filters become ultrafilters in HOD, thus answering a previously posed question.

Contribution

It demonstrates that in $ ext{P}_{ ext{max}}$ extensions of a Chang-type model, the club filter on certain cardinals restricted to HOD is an ultrafilter, addressing an open question.

Findings

Club filter on $ ext{Cof}( ext{omega})$ becomes an ultrafilter in HOD for certain cardinals.

Results hold in $ ext{P}_{ ext{max}}$ extensions of models with determinacy.

Answers an open question from Ben-Neria and Hayut (2023).

Abstract

We show that in the $\mathbb{P}_{\max}$ extension of a certain Chang-type model of determinacy, if $\kappa\in\{\omega_1, \omega_2, \omega_3\}$, then the restriction of the club filter on $\kappa\cap\mathrm{Cof}(\omega)$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [BNH23] raised by Ben-Neria and Hayut.