>>jeremy+(OP)
Here's a philosophical question that's been bothering me for a while. For large enough n, we can construct an n-state Turing machine that attempts to prove a contradiction in our current most powerful axiomatic system (let's say ZFC). We must assume that this machine loops forever, but Gödel's incompleteness theorem implies that we can't prove that.
What does this construction imply about BB(n)? In what sense is the BB sequence even well-defined if we can prove that it can't be determined?
(Edited for clarity.)