Feferman–Schütte ordinal

The \(\Gamma_0\) (pronounced "gamma-zero", also known as Feferman-Schütte ordinal) is the first ordinal inaccessible through the Veblen hierarchy. Formally, it is the first fixed point of \(\alpha \mapsto \varphi_{\alpha}(0)\), visualized as \(\varphi_{\varphi_{\varphi_{._{._..}.}(0)}(0)}(0)\) or \(\omega\ \{\{1\}\}\ \omega\) (expansion).

The growth rates of finite forms of that ordinal in different hierarchies are shown below:


 * \(f_{\Gamma_0}(n) \approx \{X,X,1,2\} \&\ n\) (fast-growing hierarchy)
 * \(H_{\Gamma_0}(n) \approx \{X,X,1,2\} \&\ n\) (Hardy hierarchy)
 * \(g_{\Gamma_0}(n) \approx \{n,n,1,2\}\) (slow-growing hierarchy)