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
The article does mention that "no Turing machine can list the Busy Beaver numbers".

>>zodiac+zf
That's clear. My point is that mathematics can't list the Busy Beaver machines.

>>codefl+Jf
> In what sense is the BB sequence even well-defined if we can prove that it can't be determined?

Every BB number can be determined. However, each one must require zillions of special cases which require their own reasoning---this is the only way to prevent the BB numbers from being computable.

So, if you consider, for example, the proof of the four-color theorem (which amounts to checking many special cases), as mathematics, then mathematics surely can determine BB(n). It's just extremely hard---as hard as doing all of mathematics (in the sense that if you could easily get BB(n) you could then take a logical proposition you are interested in proving in some axiomatic system and construct a TM to derive the statement using those axioms. Then, count how many states that TM has, and start running it. If it runs for more than BB(# states), there is no proof from your system of your proposition).