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).
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 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]
Citations
- ^ J. Van der Meeren, M. Rathjen, A. Weiermann, " An order-theoretic characterization of the Howard-Bachmann-hierarchy" (2017). Accessed 21 February 2023.
- ^ S. Feferman, " The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008." (2008), p.7. Accessed 21 February 2023.
- ^ M. Rathjen, " The Art of Ordinal Analysis" (2006), p.11. Accessed 21 February 2023.
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, 37 (2), Association for Symbolic Logic: 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.)
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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).
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 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]
Citations
- ^ J. Van der Meeren, M. Rathjen, A. Weiermann, " An order-theoretic characterization of the Howard-Bachmann-hierarchy" (2017). Accessed 21 February 2023.
- ^ S. Feferman, " The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008." (2008), p.7. Accessed 21 February 2023.
- ^ M. Rathjen, " The Art of Ordinal Analysis" (2006), p.11. Accessed 21 February 2023.
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, 37 (2), Association for Symbolic Logic: 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.)
|
| This set theory-related article is a stub. You can help Wikipedia by expanding it. |
|
| This article about a number is a stub. You can help Wikipedia by expanding it. |