A potentialist conception of ultrafinitism

TL;DR

This paper explores ultrafinitism through a potentialist lens, showing how models of finite arithmetic relate via bi-interpretability and end-extensions, offering a new perspective on ultrafinitist ideas.

Contribution

It introduces a potentialist framework for ultrafinitism, connecting models of finite arithmetic through bi-interpretability and modal end-extensions, providing a novel conceptual approach.

Findings

Models of finite arithmetic are bi-interpretable with taller models.

Ultrafinitist ideas emerge from the modal potentialist system of models.

End-extensions relate models in a way that captures ultrafinitist perspectives.

Abstract

I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that 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 model $M^+$, in which the arithmetic operations on objects taken from the original base model $M$ are totally defined in the extended world $M^+$. More generally, I explain how ultrafinitist ideas emerge in the modal potentialist system consisting of all models of arithmetic under end-extension.