Whether or not BB(n) is even defined is an interesting philosophical question.
If n is a modestly large number, it can be proven that in no consistent axiom system below a certain size can any explicit number written out in base 10 EVER be proven to be an upper bound for BB(n). If we make n something like 100 million, that encompasses all possible axiom systems that are human comprehensible.
A "finite" number with no provable upper limit - how much sense does it make to say that is well-defined? It is enough to make you take up constructivism! (Or quit math for something more sensible. Like politics.)
If n is a modestly large number, it can be proven that in no consistent axiom system below a certain size can any explicit number written out in base 10 EVER be proven to be an upper bound for BB(n). If we make n something like 100 million, that encompasses all possible axiom systems that are human comprehensible.
A "finite" number with no provable upper limit - how much sense does it make to say that is well-defined? It is enough to make you take up constructivism! (Or quit math for something more sensible. Like politics.)