Bachmann-Howard ordinal

The Bachmann-Howard ordinal is a large countable ordinal, significant for being the proof-theoretic ordinal of Kripke-Platek set theory with the axiom of infinity. It is the supremum \(\vartheta(\varepsilon_{\Omega+1})\) of \(\vartheta(\alpha)\) for all \(\alpha < \varepsilon_{\Omega+1}\) with respect to Weiermann's \(\vartheta\) and is presented as \(\psi_0(\Omega_2) = \psi_0(\psi_2(0))\) with respect to Buchholz's \(\psi\). The Bachmann-Howard ordinal has also been denoted as \(\eta_0\), however this is rare and can be confused with \(\varphi(3,0)\) in Veblen's function. It can intuitively by denoted by \(\vartheta(\Omega\uparrow\uparrow\omega)\) which may be useful for beginners, despite ordinal tetration not having a standard definition.

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