Sherwood Hachtman : ITP and SCH

Posted by Sherwood Hachtman , part of the Logic Seminar.

At

Oct. 24, 2017, 4 p.m.

In

SEO 427

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.