User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-28606698-20170415185346/@comment-28606698-20170416130719

I have some thoughts about fixed points, which are not connected with discussed notation, but I will say:

let we define recursion of Veblen function as usually

$$\varphi(\alpha, \beta)=\beta$$-th common fixed point of $$\xi=\varphi(\delta,\xi)$$ for all $$\gamma<\alpha$$

but $$\varphi(0,\beta)$$ we will define as $$\omega+\beta=\omega(1)\beta$$ instead $$\omega^\beta=\omega(3)\beta$$

then $$\varphi(\alpha,0)=\omega^\alpha$$

and for this case $$\Gamma_0$$ = first strongly critical ordinal such that $$\Gamma_0=\varphi(\Gamma_0,0)=\omega\uparrow\uparrow\omega=\omega(4)\omega=\omega(1+3)\omega.$$

Then it seems that for case $$\varphi(0,\beta)=\omega^\beta$$

$$\Gamma_0=\omega(3+3)\omega=\omega(6)\omega$$ i.e. hexation of omega. Ofcourse it's non-cannonical consideration.