Krueger showed that PFA implies that for all regular Θ ≥ ₂, there are stationarily many [H(Θ)]₁ that are internally club but not internally approachable. From countably many Mahlo cardinals, we force a model in which, for all positive n<ω and Θ ≥ ₙ₊₁, there is a stationary subset of [H(Θ)]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.
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Jakob et al. (2024) studied this question.
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