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The HOD Hypothesis and a supercompact cardinal
Author(s) -
Cheng Yong
Publication year - 2017
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600007
Subject(s) - mathematics , corollary , class (philosophy) , universality (dynamical systems) , pure mathematics , combinatorics , discrete mathematics , physics , condensed matter physics , computer science , artificial intelligence
In this paper, we prove that: if κ is supercompact and the HOD Hypothesis holds, then there is a proper class of regular cardinals in V κ which are measurable in HOD . Woodin also proved this result independently [11][W. H. Woodin, ]. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the HOD Hypothesis and supercompact cardinals, large cardinals in V are reflected to be large cardinals in HOD in a local way, and reveals the huge difference between HOD ‐supercompact cardinals and supercompact cardinals under the HOD Hypothesis.