Omitting types for finite variable fragments and complete representations of algebras

Abstract

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n≥2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is ≥2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.