Bachmann–Howard ordinal
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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. It is named after William Alvin Howard and Heinz Bachmann.
[edit] Definition
The Bachmann-Howard ordinal is defined using an ordinal collapsing function (with more details given in the relevant article):
- εα enumerates the epsilon numbers, the ordinals β with ωβ = β.
- 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 
for an extension of the Veblen functions φα to uncountable α; this extension is not completely straightforward.
[edit] References
- Bachmann, Heinz (1950), “Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen”, Vierteljschr. Naturforsch. Ges. Zürich 95: 115-147, MR0036806
- Howard, W. A. (1972), “A system of abstract constructive ordinals.”, J. Symbolic Logic 37: 355-374, MR0329869, <http://links.jstor.org/sici?sici=0022-4812%28197206%2937%3A2%3C355%3AASOACO%3E2.0.CO%3B2-Y>
- Pohlers, Wolfram (1989), Proof theory, vol. 1407, Lecture Notes in Mathematics, Berlin: Springer-Verlag, MR1026933, ISBN 3-540-51842-8
- Rathjen, Michael (August 2005). Proof Theory: Part III, Kripke-Platek Set Theory. Retrieved on 2008-04-17. (slides of a talk given at Fischbachau)
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