User blog:Fejfo/Weak fixed points

Fixed points for more functions
We all know \(\alpha) is a fixed point of \(f) if \( \alpha=f(\alpha) \). This is the basis of the Veblen hierachy, nameing the ordinals you can't name yet (using previous functions).

But not all strickly increasing functions have fixed points. Example: the succesor function doesn't have any fixed points (by definition). So I would like to define \( \alpha \) is a weak fixed point of \( f \) if \( \forall \beta < \alpha | f(\beta)<\alpha\). Using this definition the first weak fixed points of S are : \( \omega, \omega+1, \omega+2, \cdots \)

Some properties of weak fixed points
Any function from ordinals to ordinals \( f : \alpha \mapsto \beta \) has \( \beta \) as a weak fixed point because \( f(x) \in \beta \Rightarrow f(x)<\beta \). But this usually isn't the first weak fixedpoint.

I've noticed that when \( f(x)>x \) \( f^{\omega\cdot\alpha}(0) \) (transfinite itteration) the \( \alpha^\text{th} \) weak fixed point is of \( f \). This is very usefull for a general fgh: (this simple version wastes large cardinals \( f_{\Omega^\Omega}(3)= f_{\omega^\omega}(3) \) so it's not usefull for \( \Omega_\omega \))
 * \( f_0(\beta)=\beta+1 \)
 * \\( f_{\alpha+1}(\beta)=f_{\alpha}^\beta(\beta) \)
 * \( f_alpha(\beta)[n]=f_{\alpha[n]}(\beta) \) if \( \beta \not\in {\rm cf}(\alpha) \)
 * \( f_alpha(\beta)=f_{\alpha[\beta]}(\beta) \) if \( \beta \in {\rm cf}(\alpha) \)

For normal, strickly increasing and continous, functions normal fixed points are weak fixed points too. I don't think the reverse is true (all weak fixed points are normal fixed points) but I don't have a simple counter example.

Do you know any other intersing properties? Or can you prove the listed onces?