Provability with Finitely Many Variables

Abstract

For every finite $n \geq 4$ there is a logically valid sentence $\varphi_n$ with the following properties: $\varphi_n$ contains only 3 variables (each of which occurs many times); $\varphi_n$ contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): $\varphi_n$ has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that $\varphi_n$ has a proof with only n variables. To show that $\varphi_n$ has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.