User blog comment:Golapulusplex10/Halting problem as a growth rate./@comment-44657742-20200207231319

The halting problem is a problem Alan Turing proved to be unsolvable by turing machines, that by the church-turing thesis, means that it is uncomputable. The halting problem is not a growth rate.

And lastly, why does the halting problem grow faster than all levels of recursion. I don't know what you mean with this question, mainly because I don't understand levels of recursion, but I think I have an idea on what you're trying to ask. If you meant, "Why does the busy beaver grow faster than any computable function?" Then the anwser is a bit easier. The halting problem, at least on a basic level, asks if a given turing machine(or, by the Church-Turing thesis, any program) will eventually halt, or would run indefinetly. The busy beaver function is a function that asks, given a turing machine with 2 colors and n states, what's the most zeros that can be printed if the turing machine stops. This question clearly asks for the halting problem, making it uncomputable by a normal turing machine. The anwser to the original question is: As a turing machine can solve for computable functions(by definition), BB(n) will eventually dominate all computable functions.