Distinguishing Internally Club and Approachable on an Infinite Interval
Abstract
Krueger showed that PFA implies that for all regular 2, there are stationarily many [H()]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 n+1, there is a stationary subset of [H()]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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