Abstract
AbstractIn 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 . 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.
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