MAXIMALITY OF LOGIC WITHOUT IDENTITY

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( L − ωω ). In this note, we provide a fix: we show that L − ωω is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity.