User blog comment:Fejfo/Uncountable indexed veblen function/@comment-32213734-20180513045616/@comment-32213734-20180514015507

Interesting. In that article it is said that any normal function has arbitrarily large fixed points. Function f(α) is normal, if it is strictly increasing:

if α < β, then f(α) < f(β)

and continuous:

if λ is limit ordinal, then f(λ) = sup { f(α) : α < λ }

φβ(β) is not continuous: let λ = Γ0, then for β < λ supremum of φβ(β) is Γ0 ≠ φλ(λ) = φΓ 0 (Γ0) = φ(Γ0, Γ0). So, φβ(β) is not normal.