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.
Badia, G., Caicedo, X., Noguera, C. (2023). MAXIMALITY OF LOGIC WITHOUT IDENTITY. THE JOURNAL OF SYMBOLIC LOGIC, 1-16 [10.1017/jsl.2023.2].
MAXIMALITY OF LOGIC WITHOUT IDENTITY
Noguera C.
2023-01-01
Abstract
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.File | Dimensione | Formato | |
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https://hdl.handle.net/11365/1228054