Spencer Unger : The tree property and weak square

Posted by Isaac Goldbring , part of the Logic Seminar.

At

Oct. 8, 2013, 4 p.m.

In

SEO 427

Abstract

In this talk we focus on generalizing two theorems of Mitchell, but in different directions. The theorems of Mitchell are Thm 1: The tree property at $\aleph_2$ is equiconsistent with the existence of weakly compact cardinal. Thm 2: The failure of weak $\square_{\aleph_1}$ is equiconsistent with a Mahlo cardinal. We develop some of the tools needed to prove the following two theorems. Thm1': Assuming there is a weakly compact cardinal it is consistent that the tree property holds at $\aleph_2$ and the continuum is larger than $\aleph_2$. Thm2': The failure of weak $\square_{\aleph_n}$ for all n > 0 is equiconsistent with infinitely many Mahlo cardinals.