User blog comment:Fejfo/Uncountable indexed veblen function/@comment-32213734-20180513045616/@comment-32213734-20180517053825

In blog Weak fixed points it is said that a weak fixed point of a normal function is a fixed point. The opposite is also true: a fixed point of a normal function is a weak fixed point, since if α = f(α), then for β < α, since f is strictly increasing, f(β) < f(α) = α. So, derivative (f'(α) is α-th fixed point of f(α)) of normal function is equal to its weak derivative (that is α-th weak fixed point of f(α)). So, φα(β) for α < Ω is Veblen function.

Weak derivative of φΩ(β) = φβ(β) is Γ function. Proof: all Γ numbers are weak fixed points of φβ(β), since if β < Γα, then φβ(β) < Γα. Successor ordinal α + 1 is not weak fixed point of φβ(β), since φα(α) ≥ α + 1. Limit not Γ ordinal α is not weak fixed point of φβ(β), since sup(φα[n](0)) = φα(0) > α, and there is n such as φα[n](0) ≥ α, otherwise would be sup(φα[n](0)) ≤ α. And φα[n](α[n]) ≥ φα[n](0) ≥ α.

So, definition with weak fixed points does work, at least in the beginning.