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.