On the Consistency of Height and Width Potentialism

Recent work in the philosophy of set theory has furnished some arguments that height and width potentialism are inconsistent with one another. One such argument can be found in this volume (Brauer, Forth., What is forcing potentialism? Forthcoming in the Palgrave Companion to the Philosophy of Set Theory, 2024); another is forthcoming (Roberts, Ultimate v. Ms Under Review, 2024).

At the same time, others have suggested there may be some merit in the combination of height and width potentialism. Such authors have presented views that appear to manifest the combination in a non-trivial way, and have defended philosophical claims on their basis (e.g. that all sets are ultimately countable).

Clearly there is a tension here. The business of this chapter is to explain (what I take to be) its solution. I will argue that height and width potentialism are compatible, and suggest that there is hope for an attractive view in the foundations of mathematics that arises from their combination. I will do this by explaining that view and how it responds to the arguments alleging inconsistency. Along the way I will present some new results relating height and width potentialism to extensions of second order arithmetic by regularity principles.