TALK: Is it restrictive to say that there are uncountable sets?

This will be a talk in the Oslo Mathematical Logic Seminar on 17 February 2022. Handout here.

Often the view that every set is countable (call this position `countabilism’) is regarded as a viewpoint that is `restrictive’ as compared to uncountabilism (the view that there are uncountable sets). After all, doesn’t the uncountabilist hold that there are *more* sets than the countabilist? In this short talk I’ll present some results that challenge this assumption. In particular, we’ll examine a mathematical analysis of `restrictiveness’ proposed by Penelope Maddy. We’ll consider a version of Maddy-restrictiveness modified to weaker base theories that allows for countabilism, and we’ll see that in this context it is the uncountabilist, rather than the countabilist, who is making the restrictive claims. I’ll close with some philosophical remarks about (a) what we want out of set-theoretic foundations, and (b) what the role of analyses of `restrictiveness’ in this framework might be.

Talk: Title: Potentialist Sets, Intensions, and Non-Classicality

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?

Talk: Mathematical Gettier Cases

This will be a talk at Thinking about Proofs: Formal, Philosophical, Linguistical and Educational Perspectives on 27th September 2021. You can get the slides here.

Abstract: Are Gettier cases possible in mathematics? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed actual) in mathematical reasoning. By analysing these cases, I suggest that the Gettier phenomenon indicates some upshots and norms for actual mathematical practice.

Talk: Maximality and Countability (or `Some systems of set theory on which every set is countable’)

Handout here. This will be a talk at the New York Set Theory Seminar.

Abstract: It is standard in set theory to assume that Cantor’s Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor’s Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that the continuum is a proper class. We examine several theories based on maximality considerations in this framework (in particular countabilist analogues of reflection principles) and show how standard set theories (including ZFC with large cardinals added) can be incorporated. We conclude that the systems considered raise questions concerning the foundational purposes of set theory.