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An inner model is a transitive proper class substructure of the universe of sets, that satisfies $\mathsf{ZF}(\mathsf{C})$.
5
votes
Accepted
Gently changing measure
To question 1, there is such a pair. This is a minor reworking of Ashutosh's example you linked.
Start with $L,$ add an $\omega_1$-sequence of random reals $X=\langle r_{\alpha}: \alpha<\omega_1 \rang …