A Version of Predicate Logic with Two Variables That has an Incompleteness Property

In this paper, we consider predicate logic with two individual variables and general assignment models (where the set of assignments of the variables into a model is allowed to be an arbitrary subset of the usual one). We prove that there is a statement such that no general assignment model in which it is true can be finitely axiomatized. We do this by showing that the free relativized cylindric algebras of dimension two are not atomic.