A Set-Theoretic Translation of Modal Logic via Forcing

Abstract

We develop a set-theoretic translation of a normal modal extension $T_m$ of a recursively axiomatizable first-order theory $T$. We first pass to the Henkin expansion of the underlying language by adding witness constants and work with the sentence algebra of this expansion. The translation is constructed using the corresponding Lindenbaum-Tarski algebra, the Stone space of its ultrafilters, and quotient forcing by a $\sigma$-ideal of Borel sets; in particular, the meager ideal yields Cohen forcing and the null ideal yields the random forcing. In a Boolean-valued universe, we interpret the modal operator $\Box$ by membership of the Boolean value of the translated formula in a suitable filter name. We prove that this translation is sound and complete: a modal formula is provable in $T_m$ if and only if its set-theoretic translation is forced in every associated interpretation. We then build Kripke frames and models from generic extensions and show that the forcing interpretation of $\Box$ agrees with quantification over the corresponding accessibility relation. As a consequence, we obtain completeness with respect to the resulting Kripke models.