The van Benthem Characterisation Theorem for Descriptive Models
Tim Henke
Abstract:
This thesis investigate the modal and first-order model theory of the class of models over descriptive general frames. Descriptive general frames are Stone spaces with a suitable relation over which every modal logic is complete. The main theorem of this thesis is the van Benthem Characterisation Theorem for the class of descriptive general models. Moreover, a model-theoretic analysis is given to prove that many important results from classical model theory, including the Compactness Theorem for first-order logic and the upward LĂ¶wenheim-Skolem Theorem, fail on the class of descriptive general models. The main tool developed in this thesis is the descriptive unravelling, a version of the unravelling tree that is modified to be descriptive. A careful analysis of the operation is provided and three isomorphic constructions are given: a construction through duality theorems, a construction through a topological toolkit based on nets that is also developed, and an explicit construction in terms of finite and infinite paths.