Cascading Variants of Internal Approachability

Abstract

We construct models in which there are stationarily many structures that exhibit different variants of internal approachability at different levels. This answers a question of Foreman-Todorcevic. We also show that the approachability property at $\hat{I}¼$ is consistent with having a distinction between variants of internal approachability for stationarily many $N\in [H(\hat{I}{¼}^{+}){]}^{\hat{I}}¼$, answering a question of Levine.

16 pages, 0 figures