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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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Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions

I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example: can we prove under any extension …
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