A potentialist conception of ultrafinitism

2025/12/06 by Hamkins, Joel David
Mathematics · Computer Science · #Mathematical and Theoretical Analysis #Computability, Logic, AI Algorithms #Advanced Topology and Set Theory

paper · doi:10.48550/arxiv.2512.06564

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.

Citations

• Every countable model of arithmetic or set theory has a pointwise-definable end extension
• The modal logic of arithmetic potentialism and the universal algorithm
• The universal finite set
• The modal logic of set-theoretic potentialism and the potentialist\n maximality principles

Related

• Model Theory of Ultrafinitism II: Deconstructing the Term Model (First Draft)
• Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetics
• The modal logic of arithmetic potentialism and the universal algorithm
• Finitistic Properties of High Complexity
• Actual and Potential Infinity