Set theory as modal logic

Abstract

A logical systemBM ⁺ is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, □ and Δ These modalities are submitted to a finite number of axioms. □ is the usual sign of necessity, Δ corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbeddingσ of the usual theory of classesM intoBM ⁺ is constructed, such that a formulaA is provable inM if and only ifσ(A) is provable inBM ⁺. There is also an inverse imbeddingπ with an analogous property.