[Kranar]

That's a misinterpretation of what the article says. There is no actual bound in principle to what can be computed. There is a fairly practical bound which is likely BB(10) for all intents and purposes, but in principle there is no finite value of n for which BB(n) is somehow mathematically unknowable.

ZFC is not some God given axiomatic system, it just happens to be one that mathematicians in a very niche domain have settled on because almost all problems under investigation can be captured by it. Most working mathematicians don't really concern themselves with it one way or another, almost no mathematical proofs actually reference ZFC, and with respect to busy beavers, it's not at all uncommon to extend ZFC with even more powerful axioms such as large cardinality axioms in order to investigate them.

Anyhow, just want to dispel a common misconception that comes up that somehow there is a limit in principle to what the largest BB(n) is that can be computed. There are practical limits for sure, but there is no limit in principle.

[IsTom]

You can compute a number that is equal to BB(n), but you can't prove that it is the right number you are looking for. For any fixed set of axioms you'll eventually run into BB(n) too big that gets indepentent.

[Kranar]

>You can compute a number that is equal to BB(n), but you can't prove that it is the right number you are looking for.

You can't categorically declare that something is unprovable. You can simply state that within some formal theory a proposition is independent, but you can't state that a proposition is independent of all possibly formal theories.