Bachmann–Howard ordinal

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 (with the axiom of infinity) and the system CZF of constructive set theory.It is named after William Alvin Howard and Heinz Bachmann.

Definition

The Bachmann-Howard ordinal is defined using an ordinal collapsing function (with more details given in the relevant article):
*ε_α enumerates the epsilon numbers, the ordinals β with ω^β = β.
*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 $phi_{epsilon_{Omega+1(0)$ for an extension of the Veblen functions φ_α to uncountable α; this extension is not completely straightforward.

References

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