Two visions of ultrafinitism intricately intertwined
These two concepts of ultrafinitism—one positing a largest number, another asserting totality for addition and multiplication but not exponentiation—are nevertheless intricately intertwined.
We have been discussing the philosophy of ultrafinitism in an ongoing series of essays—you can find them in the ultrafinitism tag.
We mentioned that some forms of ultrafinitism assert a realm of feasibility for the natural numbers, a realm which is closed under the successor operation, as well as addition and multiplication, but it is not closed exponentiation. Another totally different approach to ultrafinitism, meanwhile, posits explicitly the existence of a largest natural number. Although these two perspectives may seem initially to be completely at odds, I should like to explain in this essay how the model-theoretic semantics of these two positions are nevertheless intricately intertwined. Indeed, when viewed from a potentialist perspective, the two ultrafinitist positions come tightly together, sharing their essential potentialist ontological commitments.
The two theories I am talking about are the theory of finite arithmetic FA, which posits the existence of a largest number, and the theory of bounded induction IΔ0, which proves the totality of addition and multiplication but not exponentiation.
The theorem I have in mind shows that these two theories are tightly connected in their model theory. Namely, we had observed previously that every model of IΔ0 has all its truncations being models of FA. What I should like to prove in this essay is the converse—every model of finite arithmetic arises as the truncation of a model of bounded induction. Indeed, every model M of finite arithmetic has a unique minimal such extension M+ in which all the additive and multiplicative arithmetic of M becomes totally defined and deterministic. Thus, the models of FA and the models of IΔ0 arrive together in their semantics—whenever you have a model M of FA, there is a unique smallest model M+ of IΔ0 of which it arises as a truncation M = M+ ↾ n, and conversely, every truncation of a model of IΔ0 has all its truncations being models of FA.
Welcome to this third in a series of critical essays on ultrafinitism, available in the ultrafinitism tag. These essays grow out of a talk I gave at the conference Ultrafinitism: Physics, Mathematics, and Philosophy held at Columbia University last April, 2025.
Let’s get into it.
