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anonymous1 comments on Second-Order Logic: The Controversy - Less Wrong
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Interesting article! What do you mean, though? Are you saying that Knuth's triple up-arrow is uncomputable (I don't see why that would be the case, but I could be wrong.)?
No, Solvent is making the same point as in my answer: in order to write down large enough numbers to guarantee that Turing machines don't halt, you need to write down a function larger than the busy beaver function, and any such function fails to be computable because you can use it to solve the halting problem.
Ah, I see now, thanks!
Basically, the busy beaver function tells us the maximum number of steps that a Turing machine with a given number of states and symbols can run for. If we know the busy beaver of, for example, 5 states and 5 symbols, then we can tell you if any 5 state 5 symbol Turing machine will eventually halt.
However, you can see why it's impossible to in general find the busy beaver function- you'd have to know which Turing machines of a given size halted, which is in general impossible.