User blog:Nayuta Ito/The attempt to make my own ordinals collapse function

Note: The first several parts may be the same as existing one. But just hold on, it will soon get crazy.

definition
I use ρ(Greek letter rho) for my OCF.

(x and y are ordinals)

$$C_0(\alpha)=\{0, x : x\leq \rho (\beta) \;\; (\beta < \alpha) \}$$

$$C_n(\alpha)=\{ x+y, xy : x,y \in C_{n-1}(\alpha) \} $$

$$C(\alpha)=\bigcup_{n<\omega} C_n(\alpha) $$

$$\rho(\alpha)=min \{ \gamma| \gamma \notin C(\alpha) \}$$

I don't know if it works, but I mean that one step in rho is "the smallest number that can't be acieved by addition and multiplication from current step."

analysis
$$\rho(0)=1$$, because the first step is zero and you can't get one from zero.

$$\rho(1)=\omega$$

$$\rho(2)=\omega^{\omega}$$

$$\rho(3)=\omega^{\omega 2}$$

$$\rho(n)=\omega^{\omega n}$$

$$\rho(\omega)=\omega^{\omega^2}$$

$$\rho(\omega+1)=\omega^{\omega^2+\omega}$$

$$\rho(\omega 2)=\omega^{\omega^2 2}$$

$$\rho(\omega^2)=\omega^{\omega^3}$$

$$\rho(\omeg^{\omega})=\omega^{\omega^{\omega}}$$

The limit is $$\epsilon_0$$.