Hardy hierarchy

The Hardy hierarchy is a certain hierarchy mapping ordinals \(\alpha\) to functions \(H_\alpha: N \rightarrow N\). For large ordinals \(\alpha\), \(H_\alpha\) grows extremely fast. The Hardy hierarchy is named after G.H. Hardy, who first described it in his 1904 paper "A theorem concerning the infinite cardinal numbers". It is less known than its cousin the fast-growing hierarchy, which grows faster. However, it can on occasion be more useful than the fast-growing hierarchy; for example, it can more easily be related to the numbers resulting from Goodstein sequences.

The functions are defined as follows:

\(\alpha[n]\) 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.
 * \(H_0(n) = n\)
 * \(H_{\alpha+1}(n) = H_\alpha(n+1)\)
 * \(f_\alpha(n) = f_{\alpha[n]}(n)\) iff \(\alpha\) is a limit ordinal