# Seminars & Colloquia Calendar

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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