. L O ] 3 N ov 2 01 1 Formalizing set theory in weak logics , searching for

We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three commuting complemented closure operators, i.e., that of diagonal-free 3-dimensional cylindric algebras (Df3’s). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multimodal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated Df3’s are not atomic and [S5,S5,S5] has Gödel’s incompleteness property. The results reported here are strong improvements of the main result of the book: Tarski, A. and Givant, S. R., Formalizing Set Theory without variables, AMS, 1987.