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Recently on this wiki has been born idea for considering an ordinal hierarchy based on catching points of FGH and SGH. We call the ordinal catching if it has the following property:

f_{\alpha}(n) < g_{\alpha}(n) < f_{\alpha}(n+1)

Let the ordinal C(\alpha) is 1+\alpha-th such ordinal.

It is known that C(0) is \psi(\Omega_\omega), but the next catching points haven't been verified.

Let's create then a hierarchy based on this function: let K_\alpha(n) = f_{C(\alpha)}(n) and consider the ordinals which are so big that SGH catches this K-hierarchy on them. Then C_1(\alpha) is 1+\alpha-th K- and SGH- catching ordinal.

By the similar pattern, we make the new hierarchy and consider catching points of SGH and that hierarchy. C_{\beta+1}(\alpha) is 1+\alpha-th catching ordinal of SGH and \beta-th hierarchy past FGH. We can also rewrite C_\beta(\alpha) as C(\beta,\alpha) for the further purposes.

Then we can make the super-hierarchy S_\alpha(n) = f_{C(0,\alpha)}(n) and the first catching point of it will be C(1,0,0).

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