User blog comment:Rgetar/Idea of program/@comment-80.98.179.160-20171224180551/@comment-28606698-20171226190734

Let's define OCF as follows:

$$C_\nu^0(\alpha)=\Omega_\nu$$

$$C_\nu^{n+1}(\alpha)=\{\beta+\gamma,\psi_\delta(\eta)|\beta,\gamma,\delta,\eta\in C_\nu^n(\alpha)\wedge\eta<\alpha\}$$

$$\psi_\nu(\alpha)=\text{min}\{\beta|\beta\notin \cup_{n=0}^\omega C_\nu^n(\alpha)\}$$

We define standard form

1) if $$\alpha=0$$ then the standard form for $$\alpha$$ is 0 and for

2) if $$\alpha$$ is not additive principal ordinal then the standard form for $$\alpha$$ is $$\sum_{i=1}^n\alpha_i$$ where each $$\alpha_i$$ is written in standard form and for all i less than n: $$\alpha_i\geq\alpha_{i+1}$$

3) if $$\alpha$$ is non-zero additive principal ordinal then the standard form for $$\alpha$$ is $$\psi_\nu(\beta)$$ where $$\beta=0$$ or $$\beta=\gamma+1$$ and $$\gamma\in C_\nu(\gamma)$$ or $$\beta$$ is a limit ordinal less than $$\text{min}\{\xi|\xi=\Omega_\xi\}$$

Then we define fundamental sequences for ordinals written in standard form:

1) if $$\alpha=\psi_\nu(beta+1)$$ then $$\alpha[n]=\psi_\nu(beta)\times n$$

I will omit other rules since I need only this for the example

If $$\alpha=\psi_0(\psi_1(\Omega_2)+1)$$ then $$\alpha=\psi_0(\beta+1)$$ and $$\beta\notin C_0(\beta)$$ so \$$alpha$$ is written not in standard form. Output of this OCF will not grow for inputs between $$\psi_1(\Omega_2)$$ and $$\Omega_2$$

According our rule $$\alpha[n]=\psi_0(\psi_1(\Omega_2))\times n$$ but it contradicts the definition of fundamental sequence as strictly increasing sequence with length $$\omega$$ whose limit is $$\alpha$$ since $$\alpha=\psi_0(\psi_1(\Omega_2)+1)=\psi_0(\psi_1(\Omega_2))$$