## FANDOM

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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 $$\vartheta(\varepsilon_{\Omega+1})$$.