Will Adkisson : The Strong Tree Property and ITP

Posted by Tom Benhamou , part of the Logic Seminar.

At

Oct. 11, 2022, 4 p.m.

In

636 SEO

Abstract

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility in much the same way that the tree property characterizes weak compactness. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph_n$ for all $1< n < \omega$ simultaneously with the strong tree property at $\aleph_{\omega+1}$. We also show that it is consistent for ITP to hold at $\aleph_n$ for all $3 < n < \omega$ and at $\aleph_{\omega+1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.