User blog comment:Fejfo/Weak fixed points/@comment-35470197-20180821073911

> Using this definition the first weak fixed points of S are : ω,ω⋅2,ω⋅3,⋯ ω,ω⋅2,ω⋅3,⋯

Thi is not true. The first weak fixed point is \(0\) by definition. Recall that when we define the notion of additive principal ordinals, we remove \(0\).

> I've noticed that when f(x)>x, fω⋅α(0) (transfinite itteration) is the αth weak fixed point is of f.

This is not true, either. You need to assume that \(f\) is strictly increasing. (It does not follow from \(f(x) > x\).

> A weak fixed point α of a normal function f is a fixed point:

This is not true, either. Your and LittlePeng9's argument is valid only when you assume the additional condition \(f(0) = 0\), which does not follow from the normality.