## Logic Seminar

Sherwood Hachtman

UIC

ITP and SCH

**Abstract:**The ineffable thin list property, ITP($\kappa$), is a tree property-like principle, introduced by Weiss to capture the combinatorial content of supercompactness. For $\kappa$ inaccessible, ITP($\kappa$) holds if and only if $\kappa$ is supercompact. On the other hand, it is consistent for ITP to hold 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, Viale has shown that the singular cardinals hypothesis (SCH) holds above $\kappa$ assuming a strengthening of ITP($\kappa$) asserting the existence of stationarily many internally unbounded guessing models. Does ITP($\kappa$) alone imply the SCH above $\kappa$? We discuss some recent results pointing towards a negative answer. This is joint work with Dima Sinapova.

Tuesday October 24, 2017 at 4:00 PM in SEO 427