User blog comment:Mh314159/The psycholog of large numbers/@comment-35470197-20190708222539/@comment-35470197-20190709011019

> But I don't understand how you can talk about the output of a function if it grows faster than all functions that are provably finite in "the usual math".

This includes a trick, "restricting to a standard natural number". Let \(P(n)\) be a formula on a natural number \(n\). Suppose that we can prove all of \(P(0),P(1),P(2),\ldots\). Then could we prove the sentence \(\forall n \in \mathbb{N}, P(n)\), i.e. "for all natural number \(n\), \(P(n)\) holds"?

Surprisingly, the answer is no. If you could prove \(\forall n \in \mathbb{N}, P(n)\), you had a proof of finite length, say \(N\). Then eliminating \(\forall n\), you could obtain proofs \(p_m\) of \(P(m)\) for all \(m \in \mathbb{N}\). The length \(L_m\) of the proof \(p_m\) would be a function on \(n\), which would not grow so fast by te construction. Then consider the case that the least length \(\ell_m\) of a proof of \(P(m)\) gives a function on \(m\) which grows very fast. Then by the observation above, there should not be a proof of \(\forall n, P(n)\). That is why the answer is no. (Actually, there are such statements \(P(n)\).)

Let \(f(n)\) be a computable function. The termination of \(f\) means the formula "for all natural number \(n\), the computation of \(f(n)\) halts within finite steps". By the reason above, the termination of \(f\) is not necessarily provable if you have proofs of the well-definedness of \(f(0),f(1),f(2),\ldots\). The transcendental integer system satisfies this property. Namely, the function \(TI(n)\) defined as "the least natural number greater than any natural numbers which are outputs of \(f(0)\) for some computable function \(f\) such that the well-definedness of \(f(0)\) is provable in \(\textrm{ZFC}\) set theory, i.e. the usual mathematics, within \(n\) letters" grows so faster that the termination of \(TI(n)\) is not provable in \(\textrm{ZFC}\) set theory. On the other hand, the well-definedness of \(TI(m)\) is provable in \(\textrm{ZFC}\) set theory for any natural number \(m\). Therefore the output of this system with a single explicit input such as \(2^{1000}\) is well-defined. Such a pointwise well-definedness is a very strong tool to create a computable large function, which I often employ.

You may ask how to formulate the provability without circular logic. It is a natural question, and is not easy to answer. It is formalised using Goedel' method in proof theory using trees of formulae. Namely, you can define a "language" as an explicit set of strings of natural numbers equipped with a syntax, "formulae in \(\textrm{ZFC}\) set theory" as strings in the language of specific type, and "proofs" as arrays of formulae satisfying finitely many restrictions. Checking whether an array of strings is a proof or not in this sense can be done by an explicit algorithm. That is why the system is actually realisable as a computable function.