Chang models over derived models with supercompact measures

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 $\hat{I}´$ that is both a limit of Woodin cardinals and a limit of $<\hat{I}´$-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 ${\mathsf{AD}}_{\mathbb{R}}+\hat{I}˜$ is regular + $\ddot{I}{‰}_1$ is $<\hat{I}{´}_{\infty }$-supercompact for some regular cardinal $\hat{I}{´}_{\infty }>\hat{I}˜$. This complements Woodin's generalized Chang model, which satisfies ${\mathsf{AD}}_{\mathbb{R}}+\ddot{I}{‰}_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.

22 pages