Bisimulations of potentialist systems

Sam Adam-Day

Abstract: A potentialist system is a first-order Kripke model based on embeddings. I define the notion of bisimulation for these systems, and provide a number of examples. Given a first-order theory T, the system Mod(T) consists of all models of T. We can then take either all embeddings, or all substructure inclusions, between these models. I show that these two ways of defining Mod(T) are bitotally bisimilar. Next, I relate the notion of bisimulation to a generalisation of the Ehrenfeucht-Fraïsé game, and use this to show the equivalence of the existence of a bisimulation with elementary equivalence with respect to an infinitary language. Finally, I consider the question of when a potentialist system is bitotally bisimilar to a system containing set-many models, providing too different sufficient conditions.