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

Up to Omega

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^{\omega}}

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

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

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

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

The limit is \epsilon_0.

Stil calm

\Omega means a fixed point and keep going on.

\rho(\Omega)=\epsilon_0

\rho(\Omega+1)=\epsilon_0^{\omega}

\rho(\Omega+\alpha)=\epsilon_0^{\omega^{\alpha}}

\rho(\Omega 2)=\epsilon_1

\rho(\Omega 3)=\epsilon_2

\rho(\Omega \omega)=\epsilon_{\omega}

\rho(\Omega^2)=\zeta_0

\rho(\Omega^2+\Omega)=\epsilon_{\zeta_0+1}

\rho(\Omega^2 2)=\zeta_1

\rho(\Omega^\Omega)=\Gamma_0

\rho(\Omega^{\Omega+\alpha})=\phi(1,\alpha,0)

\rho(\Omega^{\Omega 2})=\phi(2,0,0)

\rho(\Omega^{\Omega^2})=\phi(1,0,0,0)

\rho(\Omega^{\Omega^{\Omega}})=SVO (Is it correct?)

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