The complexity of the modal predicate logic of “true in every transitive model of ZF”

Abstract

Robert Solovay [8] investigated the version of the modal sentential calculus one gets by taking “□ϕ” to mean “ϕis true in every transitive model of Zermelo-Fraenkel set theory (ZF).” Defining aninterpretationto be a function * taking formulas of the modal sentential calculus to sentences of the language of set theory that commutes with the Boolean connectives and sets (□ϕ)* equal to the statement thatϕ* is true in every transitive model of ZF, and stipulating that a modal formulaϕisvalidif and only if, for every interpretation *,ϕ* is true in every transitive model of ZF, Solovay obtained a complete and decidable set of axioms.In this paper, we stifle the hope that we might continue Solovay's program by getting an analogous set of axioms for the modal predicate calculus. The set of valid formulas of the modal predicate calculus is not axiomatizable; indeed, it is complete.We also look at a variant notion of validity according to which a formulaϕcounts as valid if and only if, for every interpretation *,ϕ* is true. For this alternative conception of validity, we shall obtain a lower bound of complexity: every set which isin the set of sentences of the language of set theory true in the constructible universe will be 1-reducible to the set of valid modal formulas.