Cascading Variants of Internal Approachability

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 $\mu$ is consistent with having a distinction between variants of internal approachability for stationarily many $N\in[H(\mu^+)]^{\mu}$, answering a question of Levine.