Bachmann–Howard ordinal


In mathematics, the Bachmann–Howard ordinal is a large [countable ordinal].
It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory 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:
  • 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 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.
  • The Bachmann–Howard ordinal is ψ.
The Bachmann–Howard ordinal can also be defined as φεΩ+1 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.