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 \(\vartheta(\alpha)\) (using Weiermann's \(\vartheta\)) for all \(\alpha < \varepsilon_{\Omega+1}\). An early version of Bird's array notation was limited by \(\vartheta(\varepsilon_{\Omega+1})\).