The Hypothesis and a supercompact cardinal

In this paper, we prove that: if κ is supercompact and the Hypothesis holds, then there is a proper class of regular cardinals in which are measurable in . Woodin also proved this result independently 11 . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the Hypothesis and supercompact cardinals, large cardinals in are reflected to be large cardinals in in a local way, and reveals the huge difference between ‐supercompact cardinals and supercompact cardinals under the Hypothesis.