User:Vel!/Ordinal hierarchy

Observing the similarities between SGH, HH, and FGH, I propose the following definition for a generalized ordinal hierarchy:

\[f_0(n) = A(n)\] \[f_{\alpha + 1}(n) = B(n, f_\alpha)\] \[f_\alpha(n) = f_{\alpha[n]}(n)\]

for \(A: \mathbb{Z}_+ \mapsto \mathbb{Z}_+\) and \(B: \mathbb{Z}_+ \times (\mathbb{Z}_+ \mapsto \mathbb{Z}_+) \mapsto \mathbb{Z}_+\). \(\mathbb{Z}_+\) is the set of nonnegative integers. We also require that, for \(\alpha > \beta\) and sufficiently large \(n\) we have \(f_{\alpha}(n) > f_{\beta}(n)\).