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)

An early version of Bird's array notation was limited by \(\vartheta(\varepsilon_{\Omega+1})\).

The most natural known definition of the slow-growing hierarchy yields \(g_{\vartheta(\varepsilon_{\Omega+1})}(n) \approx f_{\varepsilon_0}(n)\).