Theory of pure equality

In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all (Monk 1976:240–242). This theory is consistent, as any set with the usual equality relation provides an interpretation.

The theory of pure equality was proven to be decidable by Löwenheim in 1915. If an additional axiom is added saying either that there are exactly m objects, for a fixed natural number m, or an axiom scheme is added stating there are infinitely many objects, the resulting theory is complete.