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 μ is consistent with having a distinction of variants of internal approachability for stationarily many N∈[H(μ⁺)]μ. This is obtained using a new version of Mitchell Forcing.
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Hannes Jakob (2024) studied this question.
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