Bisimulations of potentialist systems

TL;DR

This paper introduces bisimulation concepts for potentialist systems based on models and embeddings, relating them to elementary equivalence and exploring conditions for systems to be bisimilar to those with many models.

Contribution

It defines bisimulation for potentialist systems, relates it to infinitary logic, and provides conditions for systems to be bisimilar to large model systems.

Findings

Bisimulation for potentialist systems is defined and exemplified.

Bisimulation relates to elementary equivalence in infinitary logic.

Conditions are given for systems to be bisimilar to systems with many models.

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.