Bachmann–Howard ordinal

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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 Heinz Bachmann (1950) and William Alvin Howard (1972).


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 φεΩ+1(0) for an extension of the Veblen functions φα to certain functions α of ordinals; this extension was carried out by Heinz Bachmann and is not completely straightforward.[2][3]


  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.


  • 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, S2CID 44618354
  • Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 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.)