Bachmann-Howard ordinal

The \(\vartheta(\varepsilon_{\Omega+1})\) (called "Bachmann-Howard ordinal") is the first ordinal satisfying the equation \(\vartheta(\Omega^\alpha) = \alpha\). The growth rates of finite forms of that ordinal in different hierarchies are shown below:

\(f_{\(\vartheta(\varepsilon_{\Omega+1})\)}(n) \approx H(n)\) (fast-growing hierarchy, H(n) is the function that defined by Chris Bird)

\(H_{\(\vartheta(\varepsilon_{\Omega+1})\)}(n) \approx H(n)\) (Hardy hierarchy)

\(g_{\(\vartheta(\varepsilon_{\Omega+1})\)}(n) \approx X \uparrow\uparrow X \&\ n\) (slow-growing hierarchy)