Distinguishing Internally Club and Approachable on an Infinite Interval

TL;DR

This paper constructs models demonstrating the coexistence of internally club but not internally approachable sets across various cardinalities, using a new forcing technique to answer longstanding questions.

Contribution

It introduces a novel variant of Mitchell forcing to produce models with specific stationary set properties across multiple cardinal levels.

Findings

Existence of stationarily many internally club but not internally approachable sets at various cardinals.

A new forcing method enabling control over these set properties across different models.

Resolution of questions posed by Krueger regarding these set characteristics.

Abstract

Krueger showed that PFA implies that for all regular $\Theta \ge \aleph_2$, there are stationarily many $[H(\Theta)]^{\aleph_1}$ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive $n<\omega$ and $\Theta \ge \aleph_{n+1}$, there is a stationary subset of $[H(\Theta)]^{\aleph_n}$ consisting of sets that are internally club but not internally approachable. The theorem is obtained using a new variant of Mitchell forcing. This answers questions of Krueger.