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 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.
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