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 was introduced by Heinz Bachmann (1950) and William Alvin Howard (1972).

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

The Bachmann–Howard ordinal can also be defined as ${\displaystyle \phi _{\varepsilon _{\Omega +1}}(0)}$ for an extension of the Veblen functions φ_α to certain functions α of ordinals; this extension is not completely straightforward.

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