User blog comment:Edwin Shade/In Which I Try To Derive Values of Madore's OCF on My Own, and See How Far I Get/@comment-32213734-20171203100352/@comment-28606698-20171208155800

Actually the set of the rules for the mentioned collapsing function requare additional rules for cases of exponentiation of $$\Omega$$. I prefer the following collapsing function which requare minimum ammount of rules to assign fundamental sequences for all ordinals up to omega fixed point:


 * $$C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}$$,
 * $$C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}$$,
 * $$C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)$$,
 * $$\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}$$,

There and fundamental sequences are defined as follows:
 * $$\Omega_0=1$$
 * $$\Omega_{\nu+1}=\aleph_{\nu+1}$$
 * $$\psi_\nu(0)=\Omega_\nu$$
 * $$\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0$$,
 * $$\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1} \text{ and } \nu\neq 0$$
 * 1) If $$\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)$$ where $$k\geq2$$ then $$\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))$$ and $$\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])$$,
 * 2) If $$\alpha=\psi_{0}(0)=1$$, then $$\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$,
 * 3) If $$\alpha=\psi_{\nu+1}(0)$$, then $$\text{cof}(\alpha)=\Omega_{\nu+1}$$ and $$\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta$$,
 * 4) If $$\alpha=\psi_{\nu}(0)$$ and $$\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$, then $$\text{cof}(\alpha)=\text{cof}(\nu)$$ and $$\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}$$,
 * 5) If $$\alpha=\psi_{\nu}(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta$$ (and note: $$\psi_\nu(0)=\Omega_\nu$$),
 * 6) If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\eta])$$,
 * 7) If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])$$ where $$\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.$$.

I will explain the case of $$Omega^2$$ on the example of set of rules for this function:

$$\Omega_1^2=\omega^{\Omega_1+\Omega_1}=\psi_1(\psi_1(0))$$

$$\psi_0(\Omega^2)=\psi_0(\psi_1(\psi_1(0)))$$ $$\text{cof}(\psi_1(\psi_1(0)))=\text{cof}(\psi_1(0))=\psi_1(0)=\Omega_1$$ then use rule 6

$$\gamma[0]=\Omega_0=1$$

$$\gamma[1]=\psi_0(\psi_1(\psi_1(0))[\gamma[0]])=\psi_0(\psi_1(\psi_1(0)[\gamma[0]]))=$$

$$=\psi_0(\psi_1(1))=\psi_0(\omega^{\Omega_1+1})=\psi_0(\Omega\omega)$$

$$\gamma[2]=\psi_0(\psi_1(\psi_1(0))[\gamma[1]])=\psi_0(\psi_1(\psi_1(0)[\gamma[1]]))=$$

$$=\psi_0(\psi_1(\psi_0(\Omega\omega)))=\omega^{\Omega_1+\psi_0(\Omega\omega)}=\psi_0(\Omega\psi_0(\Omega\omega))$$

$$\psi_0(\Omega^2)[2]=\psi_0(\psi_1(\psi_1(0)[\gamma[2]])=\psi_0(\Omega\psi_0(\Omega\psi_0(\Omega\omega)))$$