zlacker

>>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.)

>>codefl+Ue
This kind of thing happens in math a lot. Any time you use the axiom of choice to prove something exists, it's non-constructive. It exists, but you can't get your hands on it.

I wrote a comment down below about how one could in principle determine a number which is probably BB(n), but you could never be sure. But I just had the crazy thought that if a human brain is really just an N-state Turing machine for some giant N, then any human would either wait forever or give up before finding the true BB(n) for some n. Time for bed!