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.
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.
Are you aware of the busy beaver function? Read this.
Basically, it’s impossible to write down numbers large enough for that to work.
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.