User blog comment:Ikosarakt1/Extensions of Catching function/@comment-25418284-20131119195520

Instead of focusing specifically on SGH and FGH, is it possible to develop a more general theory of ordinal hierarchies and base the catching function on that?

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 \(A: \Z_+ \mapsto \Z_+\) and \(B: \Z_+ \times (\Z_+ \mapsto \Z_+) \mapsto \Z_+\). (\(\Z_+\) is the set of nonnegative integers.)