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An inner model is a transitive proper class substructure of the universe of sets, that satisfies $\mathsf{ZF}(\mathsf{C})$.

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Can an ultrapower be undone by forcing?

For set-forcing, the answer is no, see the following article Joel David Hamkins, Greg Kirmayer, and Norman Lewis Perlmutter, Generalizations of the Kunen inconsistency, Ann. Pure Appl. Logic 163 (201 …
Joel David Hamkins's user avatar