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We show that first-order logic can be translated into a very simple and weaklogic, and thus set theory can be formalized in this weak logic. This weaklogical system is equivalent to the equational theory of Boolean algebras withthree commuting complemented closure operators, i.e., that of diagonal-free3-dimensional cylindric algebras (Df_3's). Equivalently, set theory can beformulated in propositional logic with 3 commuting S5 modalities (i.e., in themulti-modal logic [S5,S5,S5]). There are many consequences, e.g., free finitelygenerated Df_3's are not atomic and [S5,S5,S5] has G\"odel's incompletenessproperty. The results reported here are strong improvements of the main resultof the book: Tarski, A. and Givant, S. R., Formalizing Set Theory withoutvariables, AMS, 1987.

Formalizing set theory in weak logics, searching for the weakest logic with G\"odel's incompleteness property