Distinguishing Internally Club and Approachable on an Infinite Interval

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.