Large Veblen ordinal

The \(\vartheta(\Omega^\Omega)\) (called "large Veblen ordinal") is the first ordinal satisfying the equation \(\vartheta(\Omega^{\vartheta(\alpha)}) = \alpha\). The growth rates of finite forms of that ordinal in different hierarchies are shown below:

\(f_{\vartheta(\Omega^\Omega)}(n) \approx \{n,n / 2\}\) (fast-growing hierarchy)

\(H_{\vartheta(\Omega^\Omega)}(n) \approx \{n,n / 2\}\) (Hardy hierarchy)

\(g_{\vartheta(\Omega^\Omega)}(n) \approx \{n,n (0,1) 2\}\) (slow-growing hierarchy)