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Title: On Supercompactness of \omega_1
This paper studies structural consequences of supercompact- ness of ω1 under ZF. We show that the Axiom of Dependent Choice (DC) follows from “ω1 is supercompact”. “ω1 is supercompact” also implies that AD+, a strengthening of the Axiom of Determinacy (AD), is equiv- alent to ADR. It is shown that “ω1 is supercompact” does not imply AD. The most one can hope for is Suslin co-Suslin determinacy. We show that this follows from “ω1 is supercompact” and Hod Pair Capturing (HPC), an inner-model theoretic hypothesis that imposes certain smallness con- ditions on the universe of sets. “ω1 is supercompact” on its own implies that every Suslin co-Suslin set is the projection of a determined (in fact, homogenously Suslin) set. “ω1 is supercompact” also implies all sets in the Chang model have all the usual regularity properties, like Lebesgue measurability and the Baire property.  more » « less
Award ID(s):
1855757
PAR ID:
10151258
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Springer Proceedings in Mathematics & Statistics
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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