Small Veblen ordinal

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

\(\varphi(1,0)\)

\(\varphi(1,0,0)\)

\(\varphi(1,0,0,0)\)

\(\varphi(1,0,0,0,0)\)

...

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)

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