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Logic Seminar

" ISP and SCH"

Sherwood Hachtman , University of Illinois - Chicago

Location:  HILL 705
Date & time: Monday, 17 April 2017 at 5:00PM -




Time: 5:00 PM
Location: Hill 705
Abstract: ISP(\(kappa\)) is a tree property-like principle, introduced by Weiss to capture the combinatorial essence of supercompactness. For \(kappa\) inaccessible, ISP(\(kappa\)) holds if and only if \(kappa\) is supercompact. However, it is consistent relative to a supercompact that ISP holds at accessible cardinals (e.g. \(aleph_2\)) and such instances still entail some of the same consequences as supercompactness. For example, in analogy with Solovay's theorem that SCH holds above a supercompact cardinal, there is Viale's result that SCH holds above \(kappa\) assuming ISP(\(kappa\)) plus the existence of enough internally unbounded structures. Does ISP(\(kappa\)) alone imply SCH above \(kappa\)? This would follow from a positive answer to a question of Viale and Weiss: Are all \(aleph_1\)-guessing models internally unbounded? We give partial negative answers to both questions, using ideas of Sinapova and Unger.

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