Bachmann–Howard ordinal explained

In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal^[1]) 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 and .

Definition

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

• ε_α enumerates the epsilon numbers, the ordinals ε such that ω^ε = ε.
• Ω = ω₁ is the first uncountable ordinal.
• ε_Ω+1 is the first epsilon number after Ω = ε_Ω.
• 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(ε_Ω+1).

The Bachmann–Howard ordinal can also be defined as φ_εΩ+1(0) for an extension of the Veblen functions φ_α to certain functions α of ordinals; this extension was carried out by Heinz Bachmann and is not completely straightforward.^{[2] [3]}

References

• • • • Web site: Proof Theory: Part III, Kripke-Platek Set Theory . 2008-04-17 . Rathjen . Michael . August 2005 . https://web.archive.org/web/20070612112202/http://www.mathematik.uni-muenchen.de/~aehlig/EST/rathjen4.pdf . 2007-06-12 . dead. (Slides of a talk given at Fischbachau.)

Notes and References

1. J. Van der Meeren, M. Rathjen, A. Weiermann, "An order-theoretic characterization of the Howard-Bachmann-hierarchy" (2017). Accessed 21 February 2023.
2. S. Feferman, "The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008." (2008), p.7. Accessed 21 February 2023.
3. M. Rathjen, "The Art of Ordinal Analysis" (2006), p.11. Accessed 21 February 2023.