Bisimulations of potentialist systems

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 $\mathrm{Mod}(T)$ 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 $\mathrm{Mod}(T)$ are bitotally bisimilar. Next, I relate the notion of bisimulation to a generalisation of the Ehrenfeucht-Fra\"is\'e 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.