# Is an ultrafinitist way around Gödel incompleteness theorems?

I know that a similar question has been asked regarding finitism, but I'm interested in ultafinitism. That is, we define a set of numbers that has a specific upper limit. For argument's sake - let's say there are only 2 numbers: 0 and 1. So 1+1 is undefined because there is no number 2...

Does Gödel incompleteness theorem hold for these numbers? What about if the upper limit is higher - 100 or a googol?

So the question then is twofold - first, would an arithmetic limited to a sufficiently small subset of natural numbers be complete. And second, if so how high can we make that limit before it becomes incomplete?

• I think the negative reception to this question is somewhat excessive, and have upvoted to compensate. – Noah Schweber Feb 21 at 17:07
• I don't think that a specific upper limit is the right way to think about ultrafinitism. All ultrafinitists believe in two, because everybody can count that high; it's a googol that they doubt, because nobody can count that high. Still, ultrafinitists generally accept that, whenever $n$ exists, so does $n+1$. (In theory, you can prove that a googol exists using only this axiom, step by step. But in practice, this proof has about a googol steps, and you're not physically capable of producing it.) – Toby Bartels Feb 22 at 2:42

I'm not really sure what you're asking. Certainly for any natural number $$n$$, the structure $$\mathbb{N}_n$$ consisting of the natural numbers up to $$n$$ (with $$+$$ and $$\cdot$$ interpreted relationally in order to be a legitimate structure) is trivially decidable, so in that sense Godel doesn't apply to it(s theory).
But there's a huge issue here: checking whether a sentence of length $$ is true in $$\mathbb{N}_n$$ requires more than $$n$$ steps in general. So the decidability of $$\mathbb{N}_n$$ is not satisfying from an ultrafinitist standpoint, since the completeness itself isn't "justifiable within $$\mathbb{N}_n$$." Meanwhile, the non-ultrafinitist won't be impressed either since $$\mathbb{N}_n$$ is too limited a structure to treat even basic arithmetic. So this dodge doesn't seem satisfying from any perspective.
However, Godel's first incompleteness theorem still applies to these. Indeed, no theory which can prove each true quantifier-free sentence about $$\mathbb{N}$$ and, for each $$k\in\mathbb{N}$$, can prove $$Init_k:\quad\forall x(\underline{k} (where "$$\underline{m}$$" denotes the numeral corresponding to $$m$$) is going to be complete, consistent, and computably axiomatizable (see e.g this paper of Ritter). Note that such theories are not required to prove that multiplication is total, or that $$<$$ is a linear ordering of the universe, or so on: they are truly quite weak.