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. We also show that the approachability property at $\mu$ is consistent with having a distinction of variants of internal approachability for stationarily many $N\in[H(\mu^+)]^{\mu}$. This is obtained using a new version of Mitchell Forcing.