A model-theoretic approach to descriptive general frames: the van Benthem characterisation theorem
Nick Bezhanishvili, Tim Henke

Abstract:
The celebrated van Benthem characterisation theorem states that on Kripke structures modal logic is the bisimulation-invariant fragment of first-order logic. In this paper we prove an analogue of the van Benthem characterisation theorem for models based on descriptive general frames. This is an important class of general frames for which every modal logic is complete. These frames can be represented as Stone spaces equipped with a “continuous” binary relation. The proof of our theorem generalises Rosen’s proof of the van Benthem theorem for finite frames and uses as an essential technique a new notion of descriptive unravelling. We also develop a basic model theory for descriptive general frames and show that in many ways it behaves like the model theory of finite structures. In particular, we prove the failure of the Compactness theorem, of the Beth definability theorem, of the Craig interpolation theorem, and of the Upward Löwenheim-Skolem theorem.1