Feferman–Schütte ordinal

The \(\Gamma_0\) (pronounced "gamma-zero") is the first ordinal satisfying the equation \(\varphi(\alpha,0) = \alpha\). The growth rates of finite forms of that ordinal in different hierarchies are shown below:

\(f_{\varepsilon_0}(n) \approx \{X,X,1,2\} \&\ n\) (fast-growing hierarchy)

\(H_{\varepsilon_0}(n) \approx \{X,X,1,2\} \&\ n\) (Hardy hierarchy)

\(g_{\varepsilon_0}(n) = \{n,n,1,2\}\) (slow-growing hierarchy)