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.

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