Small Veblen ordinal

The small Veblen ordinal is the limit of the following sequence of ordinals \(\varphi(1, 0, 0),\varphi(1, 0, 0, 0),\varphi(1, 0, 0, 0),\ldots\) where \(\varphi\) is the extended Veblen function. Using Weiermann's theta function, it can be expressed as \(\vartheta(\Omega^\Omega)\).

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.