Relation algebra reducts of cylindric algebras and an application to proof theory

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal ca > 3 there is a logically valid sentence X, in a first-order language Y with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly ae + 1 variables, but X has no proof containing only ae variables. This solves a problem posed by Tarski and Givant in 1987. ?

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