Cascading Variants of Internal Approachability

TL;DR

This paper constructs models demonstrating the coexistence of multiple variants of internal approachability at different levels, answering a question by Foreman and using a novel Mitchell Forcing technique.

Contribution

It introduces models with stationarily many structures exhibiting various internal approachability variants and develops a new Mitchell Forcing method to achieve this.

Findings

Multiple variants of internal approachability can coexist at different levels.

The approachability property at a certain cardinal can be consistent with diverse internal variants.

A new version of Mitchell Forcing enables these consistency results.

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.