Bachmann–Howard ordinal

In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard–Bachmann 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 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_{ε_Ω+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.

Citations

References

• (Slides of a talk given at Fischbachau.)