User blog comment:Hyp cos/Catching hierarchy fails/@comment-28606698-20180116122240

We can use Hypcos's general fundamental sequences for OCFs as well as FS-systems collected on this page for ordinal indexes of functions of fast- and slow growing hierarchies. What is interesting for me how did Hypcos define what is strictly saying the growth rate comparablity between two functions?

May be

$$C_\pi(0)=\text{min}\{\alpha|\alpha=\psi_\pi(\beta)\wedge\exists n \in\mathbb N: g_\alpha(n+1)>f_\alpha(n)\}$$

$$C_\pi(\alpha+1)=\text{min}\{\beta|\beta=\psi_\pi(\gamma)\wedge\beta>C_\pi(\alpha)\wedge\exists n \in\mathbb N: g_\beta(n+1)>f_\beta(n)\}$$

$$C_\pi(\alpha)=\text{sup}\{C_\pi(\beta)|\beta<\alpha\}$$ for limit $$\alpha$$

where $$g_\alpha$$ is a function of slow growing hierarchy, $$f_\alpha$$ is a function of fast growing hierarchy, $$\psi_\pi$$ is an ordinal collapsing function for which the fundamental sequences were assigned according Hypcos's general rules