A Potentialist Conception of Ultrafinitism

I shall explore various senses in which ultrafinitism fruitfully engages with the potentialist perspective in mathematics. For example, every model M of the theory of finite arithmetic — arithmetic with a largest number, in which addition and multiplication are merely partial functions — is bi-interpretable with a strictly taller such model M⁺, in which the arithmetic of the prior numbers becomes fully defined. By iterating this construction, we find a deep connection between the models of finite arithmetic and the theory of bounded induction IΔ₀. More generally, ultrafinitist ideas emerge in the potentialist system of all models of arithmetic under end-extension.