User blog:Hyp cos/Catching Function - Better Definition

I find a better definition of catching funcion as follows. Notice that the definition is all made from ordinals and sets - nothing about FGH or SGH. Then it has these properties.
 * \(B(0,\beta)=\beta+1\)
 * \(A(\alpha,\beta,0)=\beta\)
 * \(A(\alpha,\beta,\gamma+1)=B(\alpha,A(\alpha,\beta,\gamma))\)
 * \(A(\alpha,\beta,\gamma)=sup\{A(\alpha,\beta,\gamma[n]):n<\omega\}\) iff \(\gamma\) is a limit.
 * \(B(\alpha+1,\beta)=A(\alpha,\beta,\beta)\)
 * \(B(\alpha,\beta)=sup\{B(\alpha[n],\beta):n<\omega\}\) iff \(\alpha\) is a limit.
 * The set \(S=\{\alpha:B(\alpha,\omega)=\alpha\}\)
 * \(C(0)=minS\)
 * \(C(\alpha+1)=min\{\beta:\beta\in S\land\beta>\alpha\}\)
 * \(C(\alpha)=sup\{C(\alpha[n]):n<\omega\}\) iff \(\alpha\) is a limit.
 * 1) \(B(\alpha,n)=g_{B(\alpha,n)}(m)=f_{\alpha}(n)\).
 * 2) The growth rate of \(g_{B(\alpha,\omega)}(n)\) in FGH is \(\alpha\).
 * 3) \(\alpha\in S\) iff \(f_{\alpha}(n)\) is comparable to \(g_{\alpha}(n)\), in another word, exist k, for any n, \(f_{\alpha}(n)<g_{\alpha}(n+k)\).