This will be a talk for the Philosophy Department Colloquium at Ohio University in Athens, OH on April 30th, 2026. I am very grateful for the invitation.
A potentialist perspective on ultrafinitism, Ohio University
Abstract. Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist. I shall give an account of the deep model-theoretic connections between two otherwise very different-seeming approaches to ultrafinitism, which differ on the question of whether the feasible numbers are closed under successor. These connections are revealed and strengthened by adopting a potentialist outlook on the nature of arithmetic, where one’s realm of feasibility can be successively enlarged and enlarged again. This approach opens the door to a modal perspective on arithmetic and the idea of expressing core ultrafinitist principles in a modal vocabulary. Ultimately, this is an actualist modal model theory of ultrafinitist potentialism, which I take to shed light on the nature of ultrafinitism.
See also:
- Recent essay series on Infinity More
- Joel David Hamkins, “A potentialist conception of ultrafinitism,” to appear in Philosophia Mathematica, 2026, arxiv:2512.06564.