Chang models over derived models with supercompact measures

TL;DR

This paper constructs a Chang-type model with supercompact measures over a derived model of a hod mouse, demonstrating consistency relative to a Woodin cardinal and exploring its set-theoretic properties.

Contribution

It introduces a new Chang-type model with supercompact measures extending derived models of hod mice, advancing understanding of large cardinals and determinacy.

Findings

The model satisfies AD_R and Θ is regular.

ω_1 is <δ_∞-supercompact in the model.

Consistency is shown relative to a Woodin cardinal.

Abstract

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $\delta$ that is both a limit of Woodin cardinals and a limit of ${<}\delta$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + \Theta$ is regular + $\omega_1$ is ${<}\delta_{\infty}$-supercompact for some regular cardinal $\delta_{\infty}>\Theta$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+\omega_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.