Bachmann-Howard ordinal

The Bachmann-Howard ordinal is a large ordinal, significant for being the proof-theoretic ordinal of Kripke-Platek set theory with the axiom of infinity. It is the supremum of the range of Weiermann's \(\vartheta\) function, and since the domain of the function is \(\varepsilon_{\Omega+1}\), it is typically denoted \(\vartheta(\varepsilon_{\Omega+1})\) (which is not strictly correct but still unambiguous).

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 a function 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 \(g_{\vartheta(\varepsilon_{\Omega+1})}(n) \approx f_{\varepsilon_0}(n)\).