User blog comment:P進大好きbot/What does a computable large number mean?/@comment-4224897-20180610135217/@comment-27513631-20180612000331

Mind if I mention that there kinda is a hard bound on computable numbers, if viewed from a definitional sense?

Namely, fix a theory \(T\) (language \(L\)) which contains a sort \(\mathbb N\) and the semiring operations. Now, a number \(n\) (defined in \(L\)) is computable (relative to \(T\)) if in all models of \(T\), \(n\) is actually finite. Note that TREE(3) is computable relative to PA (compute it), and BB(4) is too (value is provable), but BB(2000) isn't (halting is independent and hence value is too).

Generally, choice of \(T\) isn't very (but can be somewhat) relevant, so it makes sense to fix a canonical (often weak) \(T\) for each \(L\) of interest (note that set theory would be two-sorted in this context, with an additional sort for \(\mathbb N\) which is isomorphic to \(\omega\)).