| The ultimate generalization is to use an inductive definition based on recursive ideals. For instance the fast growing hierarchy f_alpha: https://en.wikipedia.org/wiki/Fast-growing_hierarchy 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. |