User blog comment:Ikosarakt1/Extensions of Catching function/@comment-25418284-20131119195520/@comment-25418284-20131120013552

@Penguin Good idea.

@Ikosarakt Here are the definitions of SGH, HH, and FGH superimposed:

\[f_0(n) = 0,n,n + 1\] \[f_{\alpha + 1}(n) = f_\alpha(n) + 1,f_\alpha(n + 1),f_\alpha^n(n)\] \[f_{\alpha}(n) = f_{\alpha[n]}(n)\]

Notice that for the first equation, each right hand side is a function of \(n\); for the second equation, each right hand side is function of \(f_\alpha\) and \(n\). We call the right hand side of the first equation \(A(n)\), and the right hand side of the second \(B(f_\alpha, n)\). Thus all we need to define an ordinal hierarchy is two functions \(A,B\), one that specifies a base rule and the other a prime rule.

I will note that maybe we can eliminate \(A\) from all this by setting \(A(n) = n + 1\). This is \(f_0\) in FGH, \(f_1\) in HH, and \(f_{\omega + 1}\) in SGH &mdash; I suspect these offsets have minimal effect.