Bachmann–Howard ordinal

Definition

• 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 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.