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The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...

This is a spinoff of this earlier question of mine. Short version: What measures in $L(\mathbb{R})$ can be gotten from "potentially club" filters, under appropriate hypotheses? Long version: ...

This question was previously asked and bountied at MSE, unsuccessfully. I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...