A Potentialist Conception of Ultrafinitism

Abstract 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^{+} $, in which the arithmetic of the prior numbers becomes fully defined. By iterating this construction, we find a deep connection between the models of finite arithmetic and the theory of bounded induction I$ \Delta_{0} $. More generally, ultrafinitist ideas emerge in the potentialist system of all models of arithmetic under end-extension.