The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal

TL;DR

This paper demonstrates that assuming a supercompact cardinal and the $ ext{HOD}$ Hypothesis, there are many measurable cardinals in $ ext{HOD}$ below $ ext{V}_ ext{κ}$, highlighting differences between $ ext{HOD}$-supercompact and supercompact cardinals.

Contribution

It proves a new result linking supercompact cardinals and the $ ext{HOD}$ Hypothesis, and establishes Woodin's Local Universality Theorem as a corollary.

Findings

Existence of a proper class of measurable cardinals in $ ext{HOD}$ below $ ext{V}_ ext{κ}$

Reflection of large cardinals from $ ext{V}$ to $ ext{HOD}$ under the $ ext{HOD}$ Hypothesis

Large differences between $ ext{HOD}$-supercompact and supercompact cardinals under the hypothesis

Abstract

In this paper, we prove that: if $\kappa$ is supercompact and the $\mathsf{HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_{\kappa}$ which are measurable in $\mathsf{HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\mathsf{HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\mathsf{V}$ are reflected to be large cardinals in $\mathsf{HOD}$ in a local way, and reveals the huge difference between $\mathsf{HOD}$-supercompact cardinals and supercompact cardinals under the $\mathsf{HOD}$ Hypothesis.