Truth of Fundamental Theorem of Arithmetic beyond some large number

Question

Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the universe. Can a Prime factorization of $n$ still be possible? Does it enter the realms of philosophy or is it still a tangible mathematical concept?

Answer 1

This position in philosophy of mathematics is known as ultrafinitism. Ultrafinitists do not accept the existence of arbitrarily large natural numbers. Among other things, I think the claim is that when we assume mathematical induction (which is basically all that is needed to prove unique prime factorization) we are making an unjustified assumption about how large numbers work based on our experiences with how small numbers work. An ultrafinitist might argue that there's no way to really know that this assumption is valid.

I should make it clear that ultrafinitism is something of a fringe position. Many mathematicians are happy with induction and in fact with much stronger assumptions like the axiom of choice. I myself am mildly agnostic about whether computations that can't be performed in the physical universe can reasonably be said to have outputs, but most of the time when I do mathematics I don't think about this kind of stuff.

Answer 2

Yes, there exists a unique prime factorization.

No, we probably won't ever know what it is.