Seminars & Colloquia Calendar
" 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. |
