# Incompactness of the second uncountable cardinal

In a celebrated paper from 1997, Shelah proved that Pr1(w2,w2,w2,w) is a theorem of ZFC, and it remains open ever since whether moreover Pr1(w2,w2,w2,w1) holds.

In an unpublished note from 2017, Todorcevic proved that a certain weakening of the latter follows from CH.

In a recent paper with Zhang (arXiv:2104.15031), we gave a few weak sufficient conditions for Pr1(w2,w2,w2,w1) to hold.

In an even more recent paper (arXiv:1910.02419v2), Shelah proved that it holds, assuming the existence of a nonreflecting stationary subset of S^2_0, hence, the strength of the failure is at least that of a Mahlo cardinal.

In this talk, we'll prove that (weak forms of) square(w2) are sufficient for Pr1(w2,w2,w2,w1) to hold, thus raising the strength of the failure to that of a weakly compact cardinal.

This is joint work with Jing Zhang.

Last Updated Date : 27/07/2021