You can read a fuller description of the project outline here.

Abstract: Set-theoretic potentialism is the view that whatever sets there are, there could always be more. Standard formulations of set-theoretic potentialism are given a modal two-valued treatment. We use some variety of modal logic — be it standard, positive-free, or negative-free — with an underlying two-valued classical logic. In this sense, the approach is local in that at the core of the semantics we look at what is made true by individual worlds. Mirroring theorems can then be used to recover much of standard set theory from a more global perspective, but unrestricted quantifiers are understood as implicitly modal (with the universal and existential prefixed by a necessity and possibility operator respectively).

The situation is complicated by a recent division between strict and liberal potentialism. Roughly speaking, liberal potentialists think that the modal facts about how the universe unfolds are fixed from the off. Strict potentialists, by contrast, hold that new facts can be “made true” as new sets come into being.

An further complication is that when working with collections, it is important to distinguish between two kinds. For extensional entities it is sufficient for two such collections to be identical that they have exactly the same members. Intensional entities on the other hand can change members across different possible worlds, and just because two classes have the same members over a domain does not make them the same class.

This project aims to (1.) establish a framework for dealing with intensions, and (2.) examine the strength of theories that one gets out. In particular, by using features of the possible world structure, we will provide an algebraic semantics to assess these questions. On this basis, there are at least preliminary indications that:

(1.) Though we might ascribe a many-valued logic to liberal potentialist, they should still hold on to classical logic globally.

(2.) The strict potentialist should accept intuitionistic logic. However, the strength of the logic motivated is contingent upon both the expressive resources allowed and the variety of potentialism in question.