This will be a talk in the Oslo Infinity and Intensionality project seminar on 24th November 2021. Handout available here.
Title: Potentialist Sets, Intensions, and Non-Classicality
Abstract: There’s lots of different kinds of set-theoretic potentialism. We could add ranks, or subsets, or do something with a little of both. At the same time, we could be liberal potentialists (thinking that the modal truths are fixed) or strict potentialist (thinking that the modal truths get `made true’ as more sets come into being). The situation with sets is now a lot better understood on all these combinations, we have decent semantics for both strict and liberal potentialists and accounts of mirroring theorems. But what about classes (conceived of as intensional entities)? These are collection-like entities that can change members as more sets come to exist. Examples include the class of all (self-identical) sets, the class of all countable sets, and many more besides. So how do we handle talk of these intensional entities on each kind of potentialism?
In this talk I’ll do the following:
1. Give a very rough description of some in progress work on how to handle intensions, with truth values being collections of worlds capturing *when* a set gets into an intensional class.
2. Consider some philosophical questions regarding the approach (this will be the bulk of the talk). In particular we’ll discuss:
2.i. Is this kind of project anti-potentialist?
2.ii. What intensions exist?
2.iii What motivations might there be for strict potentialism in this context?