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In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by Heinz Bachmann (1950) and William Alvin Howard (1972).
Definition

The Bachmann–Howard ordinal is defined using an ordinal collapsing function:

• εα enumerates the epsilon numbers, the ordinals ε such that ωε = ε.
• Ω = ω1 is the first uncountable ordinal.
• εΩ+1 is the first epsilon number after Ω = εΩ.
• ψ(α) is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying ordinal addition, multiplication and exponentiation, and ψ to previously constructed ordinals (except that ψ can only be applied to arguments less than α, to ensure that it is well defined).
• The Bachmann–Howard ordinal is ψ(εΩ+1).

The Bachmann–Howard ordinal can also be defined as $$\phi _{{\varepsilon _{{\Omega +1}}}}(0)$$ for an extension of the Veblen functions φα to certain functions α of ordinals; this extension is not completely straightforward.
References

Bachmann, Heinz (1950), "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen", Vierteljschr. Naturforsch. Ges. Zürich, 95: 115–147, MR 0036806
Howard, W. A. (1972), "A system of abstract constructive ordinals.", Journal of Symbolic Logic, Association for Symbolic Logic, 37 (2): 355–374, doi:10.2307/2272979, JSTOR 2272979, MR 0329869
Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 3-540-51842-8, MR 1026933
Rathjen, Michael (August 2005). "Proof Theory: Part III, Kripke-Platek Set Theory" (PDF). Archived from the original (PDF) on 2007-06-12. Retrieved 2008-04-17. (Slides of a talk given at Fischbachau.)