The Hypothesis and a supercompact cardinal

Abstract

In this paper, we prove that: if κ is supercompact and the boldHOD Hypothesis holds, then there is a proper class of regular cardinals in Vκ which are measurable in boldHOD. Woodin also proved this result independently . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the boldHOD Hypothesis and supercompact cardinals, large cardinals in boldV are reflected to be large cardinals in boldHOD in a local way, and reveals the huge difference between boldHOD‐supercompact cardinals and supercompact cardinals under the boldHOD Hypothesis.