Mathematics – Logic
Scientific paper
2011-11-03
Mathematics
Logic
24 pages, 3 figures. First version, additional material in http://www.renyi.hu/~nemeti/FormalizingST.htm
Scientific paper
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 (Df_3's). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multi-modal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated Df_3's are not atomic and [S5,S5,S5] has G\"odel'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.
Andreka Hajnal
Nemeti Istvan
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