User blog comment:Denis Maksudov/Slowly growing ordinal function and FS up to BHO./@comment-24920136-20170404223258/@comment-28606698-20170405080941

As I understand you proposed to use definition of OCF with

$$C_0=\{0,\Omega\}$$

$$C_{n+1}(\alpha)=\{\beta+\gamma, \varphi(\beta,\gamma), \psi(\eta)|\beta,\gamma,\eta\in C_n(\alpha);\eta<\alpha\}$$

$$C(\alpha)=\cup_{n=0}^\omega C_n(\alpha)$$

$$\psi(\alpha)=\text{min}\{\beta|\beta\notin C(\alpha)\}$$

I never tested how such OCF works but intuitively I guess

$$\psi(0)=\Gamma_0$$

$$\psi(\alpha)=\Gamma_\alpha$$

$$\psi(\Omega)=\Gamma_{\Gamma_{...}}=\varphi(1,1,0)=\theta(\Omega+1,0)=\psi'(\Omega^{\Omega+1})$$.

Then my $$\psi'$$-function, this $$\psi$$-function and this $$\theta$$-function catche one another near $$\theta(\varepsilon_{\Omega+1},0)$$

But if we use closure operations, which are including the Veblen function, then to assign FS for all limit ordinals, we should to add five rules for Veblen function in general system of rules for FS. I dream to obtain definition for OCF such that

$$\psi'(\Omega_{k+1})=\psi'(\Omega_{k}\uparrow^2 \omega)$$