Embedding sheaf models for set theory into Boolean-valued permutation models with an interior operator (Q1096625)

For any Boolean algebra B with an interior operator and an action by the Freyd permutation group [cf. \textit{P. Freyd}, J. Pure Appl. Algebra 46, 49-58 (1987; Zbl 0611.18003)], any sentence in epistemic set theory ZFE is provided with a truth value in B (via a construction of a Boolean valued permutation model). It is shown that to each Grothendieck topos E, there is such a Boolean algebra B with the property that the sentences of intuitionistic set theory ZFI which are valid in the (Fourman-) interpretation in E are exactly those which (by translation into epistemic set theory) acquire truth value 1 in B.