Strict potentialism in modal mirrors

TL;DR

This paper explores strict potentialism through bimodal logic, using mirroring theorems to simplify theories and applying the framework to set theory and Cantorian sets.

Contribution

It introduces a bimodal logic framework for strict potentialism and simplifies theories via mirroring theorems, linking to set theory and foundational principles.

Findings

Switching off object generation yields a restricted plural logic.

Switching off truth determination yields an intuitionistic logic.

Applications include Weyl-inspired set theory and Cantor's domain principle.

Abstract

Potentialism is the view that objects are successively generated in an incompletable process. A strict version of the view adds that truths are successively determined. Strict potentialism can be analyzed using two modalities: one for the generation of objects, another for truths becoming determined. The result is a classical bimodal logic. We obtain simpler and more user-friendly theories by invoking so-called mirroring theorems to ``switch off'' one or both modalities, in return for a less classical logic. When the modality of object generation is switched off, we obtain a restricted plural logic. When the modality of truth determination is switched off, the logic becomes intuitionistic. Finally, the value of this general approach to strict potentialism is illustrated by applications to a Weyl-inspired predicative set theory, Cantor's domain principle, and strict potentialism about Cantorian sets.