Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to [Wikipedia](https://en.wikipedia.org/wiki/Ultrafinitism), it has been primarily studied by Alexander Esenin-Volpin. On his [opinions page](https://sites.math.rutgers.edu/%7Ezeilberg/OPINIONS.html), Doron Zeilberger has often expressed similar opinions.


Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get your hands on coming from a purely classical perspective?


Edit: Neel Krishnaswami in his answer gave a link to a paper by Vladimir Sazonov ([On Feasible Numbers](https://doi.org/10.1007/3-540-60178-3_78)) that seems to go a ways towards giving a formal foundation to ultrafinitism.


First, Sazonov references a result of Parikh's which says that Peano Arithmetic can be consistently extended with a set variable $F$ and axioms $0\in F$, $1\in F$, $F$ is closed under $+$ and $\times$, and $N\notin F$, where $N$ is an exponential tower of $2^{1000}$ twos.


Then, he gives his own theory, wherein there is no cut rule and proofs that are too long are disallowed, and shows that the axiom $\forall x\ \log \log x < 10$ is consistent.



Is there any formal foundation to ultrafinitism?

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> Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get your hands on coming from a purely classical perspective?
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There are no foundations for ultrafinitism as satisfactory for it as (say) intuitionistic logic is for constructivism. The reason is that the question of what logic is appropriate for ultrafinitism is still an open one, for not one but several different reasons.


First, from a traditional perspective -- whether classical or intuitionistic -- classical logic is the appropriate logic for finite collections (but not K-finite). The idea is that a finite collection is surveyable: we can enumerate and look at each element of any finite collection in finite time. (For example, the elementary topos of finite sets is Boolean.) However, this is not faithful to the ultra-intuitionist idea that a sufficiently large set is *impractical* to survey.


So it shouldn't be surprising that more-or-less ultrafinitist logics arise from complexity theory, which identifies "practical" with "polynomial time". I know two strands of work on this. The first is Buss's work on $S^1\_2$, which is a weakening of Peano arithmetic with a weaker induction principle:


$$A(0) \land (\forall x.\;A(x/2) \Rightarrow A(x)) \Rightarrow \forall x.\;A(x)$$


Then any proof of a forall-exists statement has to be realized by a polynomial time computable function. There is a line of work on bounded set theories, which I am not very familiar with, based on Buss's logic.


The second is a descendant of Bellantoni and Cook's work on programming languages for polynomial time, and Girard's work on linear logic. The Curry-Howard correspondence takes functional languages, and maps them to logical systems, with types going to propositions, terms going to proofs, and evaluation going to proof normalization. So the complexity of a functional program corresponds in some sense to the practicality of cut-elimination for a logic.


IIRC, Girard subsequently showed that for a suitable version of affine logic, cut-elimination can be shown to take polynomial time. Similarly, you can build set theories on top of affine logic. For example, Kazushige Terui has since described a set theory, [Light Affine Set Theory](https://doi.org/10.1023/B:STUD.0000034183.33333.6f), whose ambient logic is linear logic, and in which the provably total functions are exactly the polytime functions. (Note that this means that for Peano numerals, multiplication is total but exponentiation is not --- so Peano and binary numerals are not isomorphic!)


The reason these proof-theoretic questions arise, is that part of the reason that the ultra-intuitionist conception of the numerals makes sense, is precisely because they deny large *proofs*. If you deny that large integers exist, then a proof that they exist, which is larger than the biggest number you accept, doesn't count! I enjoyed Vladimir Sazonov's paper ["On Feasible Numbers"](https://doi.org/10.1007/3-540-60178-3_78), which explicitly studies the connection.


I should add that I am not a specialist in this area, and what I've written is just the fruits of my interest in the subject -- I have almost certainly overlooked important work, for which I apologize.


