Slow-growing hierarchy

The slow-growing hierarchy is a certain hierarchy mapping ordinals \(\alpha\) to functions \(H_\alpha: N \rightarrow N\). Like its name suggests, it grows much slower than its cousins the Fast-growing hierarchy and the Hardy hierarchy. It is less well known than the fast-growing hierarchy, but on occasion it can be useful. Since it grows the slowest of all the ordinal hierarchies, it may be the best suited to stratify the growth rates of functions.

The functions are defined as follows:


 * \(G_0(n) = 0\)
 * \(G_{\alpha+1}(n) = H_\alpha(n)+1\)
 * \(G_\alpha(n) = G_{\alpha[n]}(n)\) iff \(\alpha\) is a limit ordinal

\(\alpha[n]\) denotes the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\). Definitions of \(\alpha[n]\) can vary, giving different verisons of the Hardy hierarchy.