Who Can Name the Bigger Number?
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.<p>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?<p>(Edited for clarity.)