A Finite First-Order Theory of Classes

We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. These axioms have the property of being easily orientable into rewrite rules. This allows us to give finite first-order axiomatizations of arithmetic and real fields theory, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework.

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