User blog:Ikosarakt1/Extensions of Catching function

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 there is a constant relation between arbitrary n and m:

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

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.