Small Veblen ordinal

The \(\vartheta(\Omega^\omega)\) (called "small Veblen ordinal") is the limit ordinal of sequence of ordinals:

\[\varphi(1,0,0) = \vartheta(\Omega) = \Gamma_0\]

\[\varphi(1,0,0,0) = \vartheta(\Omega^2)\]

\[\varphi(1,0,0,0,0) = \vartheta(\Omega^3)\]

\[\varphi(1,0,0,0,0,0) = \vartheta(\Omega^4)\]

\[\ldots\]

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

\(f_{\vartheta(\Omega^\omega)}(n) \approx \{X,X (1) 2\} \&\ n\) (fast-growing hierarchy)

\(H_{\vartheta(\Omega^\omega)}(n) \approx \{X,X (1) 2\} \&\ n\) (Hardy hierarchy)

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

Harvey Friedman's tree(n) function grows as fast as \(f_{\vartheta(\Omega^\omega)}(n)\) in the fast-growing hierarchy.