2026/07/28 by Santiago Jockwich, Sourav Tarafder, Giorgio Venturi
Mathematics · #math.LO
We connect modal set theory with Boolean-valued models by developing an internal Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra B, we view its elements as ``local perspectives on truth'' inside the Boolean-valued universe V(B) and interpret the modal operators using an accessibility relation R on B defined by co-consistency (equivalently, Boolean compatibility): aRb iff a\wedge b≠ 0. Our central conceptual point is that, for set-theoretic sentences p, the internal modality \Diamond p holds at b iff there is an ultrafilter U of B containing b such that the classical quotient V(B)/U satisfies p. We compute several general and algebra-dependent modal validities, and analyze the special behavior of complete atomic Boolean algebras. Finally, adopting a translation-based semantics on the nonzero part B+=B∖\0\, we prove a soundness-and-completeness theorem: the normal logic \KTB is exactly the set of modal formulas valid in all translated co-consistency models with parameters.