Since ordinals are the ultimate recursion tool, any such function will ultimately grow faster than whatever recursive definition you can cook up by hand.
For instance even the slow growing hierarchy g_alpha, which grows very very slowly catches up to the fast hierarchy at extremely large ordinals.
Some examples:
- f_0 behaves like addition, f_1 like multiplication, f_2 like exponentiation, f_3 like tetration, f_w like Ackerman
- Graham number is bounded by f_{w+1}(64)
- Goodstein function behaves like f_{epsilon_0} (this show how much faster it grows than Ackerman). Similar for the Kirby-Paris Hydra
- n-Iterated goodstein behaves live f_{\phi(n-2,0)} where \phi is Veblen's function
- the tree function from Kruskal's tree theorem
behaves like f_{\psi(Omega^Omega^omega}, ie by the small Veblen ordinal. And the graph function from the Robertson-Seymour theorem behaves like \psi(Omega_omega).
By contrast the slow growing hierarchy grows extremely slowly, for instance g_{epsilon_0} only grows like tetration. But it still catches up to the fast growing hierarchy at very large ordinals.
Like I said, I can't really follow this stuff, but maybe the ultimate would be the Busy Beaver function? Is there such a thing as a proof of how many functions can be between something computable and Busy Beaver, given some sort of constraints?
Yes, the Busy Beaver function grows faster than the functions I gave, basically it behaves like f_{w^1_CK} (although this does not really make sense because w^1_CK, the Church Kleen ordinal, is not recursive).
By "ultimate generalisation" I was speaking about a family of computable functions that ultimately grow faster than other functions you could cook up by hand: you put all the recursivity in the ordinals, they are there for that, and logicians are very good at constructing very large (recursive) ordinals.
If you just want a very very fast growing function (but that is still computable, unlike busy beaver), you can do the a light busy beaver version: Beaver(N)=the maximal output of all turing machines T with N state where there is a proof in ZFC of length <=N that T terminates.
This function grows much faster than the other functions I described (basically it is a f_{ordinal consistency of ZFC}.
If you are interested in this kind of ideas, there is this reddit post that develops these ideas:
https://www.reddit.com/r/math/comments/283298/how_to_compute...
Since ordinals are the ultimate recursion tool, any such function will ultimately grow faster than whatever recursive definition you can cook up by hand.
For instance even the slow growing hierarchy g_alpha, which grows very very slowly catches up to the fast hierarchy at extremely large ordinals.
Some examples:
- f_0 behaves like addition, f_1 like multiplication, f_2 like exponentiation, f_3 like tetration, f_w like Ackerman
- Graham number is bounded by f_{w+1}(64)
- Goodstein function behaves like f_{epsilon_0} (this show how much faster it grows than Ackerman). Similar for the Kirby-Paris Hydra
- n-Iterated goodstein behaves live f_{\phi(n-2,0)} where \phi is Veblen's function
- the tree function from Kruskal's tree theorem behaves like f_{\psi(Omega^Omega^omega}, ie by the small Veblen ordinal. And the graph function from the Robertson-Seymour theorem behaves like \psi(Omega_omega).
By contrast the slow growing hierarchy grows extremely slowly, for instance g_{epsilon_0} only grows like tetration. But it still catches up to the fast growing hierarchy at very large ordinals.