User blog:Edwin Shade/In Which I Try To Derive Values of Madore's OCF on My Own, and See How Far I Get

Examples I
$$\psi(0)=\epsilon_0$$


 * FS:$$\{\omega,\omega^{\omega},\omega^{\omega^{\omega}},\omega^{\omega^{\omega^{\omega}}},\omega^{\omega^{\omega^{\omega^{\omega}}}},\ldots\}$$

$$\psi(1)=\epsilon_1$$

$$\psi(2)=\epsilon_2$$

$$\psi(3)=\epsilon_3$$

$$\psi(4)=\epsilon_4$$

$$\psi(\omega)=\epsilon_{\omega}$$


 * FS:$$\{\psi(0),\psi(1),\psi(2),\psi(3),\psi(4),\ldots\}$$

$$\psi(\omega+1)=\epsilon_{\omega+1}$$

$$\psi(\epsilon_0)=\epsilon_{\epsilon_0}$$

$$\psi(\psi(\psi(0)))=\epsilon_{\epsilon_{\epsilon_0}}$$

$$\psi(\psi(\psi(\psi(0))))=\epsilon_{\epsilon_{\epsilon_{\epsilon_0}}}$$

Generalized Rules
$$\psi(\gamma)=\epsilon_{\gamma}$$, where $$\gamma$$ is a previously constructed ordinal.

$$\psi(\alpha)[n]=\psi(\alpha)\uparrow\uparrow n$$ if $$\alpha$$ is a successor ordinal.

$$\psi(\beta)[n]=\psi(\beta[n])$$ if $$\beta$$ is a limit ordinal.

Examples II
$$\psi(\Omega)=\zeta_0$$


 * FS:$$\{\psi(0),\psi(\psi(0)),\psi(\psi(\psi(0))),\psi(\psi(\psi(\psi(0))))\psi(\psi(\psi(\psi(\psi(0))))),\ldots\}$$

$$\psi(\Omega+1)=\epsilon_{\zeta_0+1}$$

$$\psi(\Omega+2)=\epsilon_{\zeta_0+2}$$

$$\psi(\Omega+3)=\epsilon_{\zeta_0+3}$$

$$\psi(\Omega+\omega)=\epsilon_{\zeta_0+\omega}$$


 * FS:$$\{\psi(\Omega),\psi(\Omega+1),\psi(\Omega+2),\psi(\Omega+3),\psi(\Omega+4),\ldots\}$$

$$\psi(\Omega+\psi(0))=\epsilon_{\zeta_0+\epsilon_0}$$

$$\psi(\Omega+\psi(\Omega))=\epsilon_{\zeta_0+\zeta_0}=\epsilon_{{\zeta_0}2}$$


 * FS:$$\{\psi(\Omega+\psi(0)),\psi(\Omega+\psi(\psi(0))),\psi(\Omega+\psi(\psi(\psi(0)))),\psi(\Omega+\psi(\psi(\psi(\psi(0))))),\psi(\Omega+\psi(\psi(\psi(\psi(\psi(0)))))),\ldots\}$$

$$\psi(\Omega+\epsilon_{\zeta_0+1})=\epsilon_{\epsilon_{\zeta_0+1}}$$

$$\psi(\Omega+\epsilon_{\epsilon_{\zeta_0+1}})=\epsilon_{\epsilon_{\epsilon_{\zeta_0+1}}}$$

Generalized Rule
$$\psi(\Omega+\gamma)=\epsilon_{\zeta_0+\gamma}$$, where $$\gamma$$ is a previously constructed ordinal.

$$\psi(\Omega+\alpha)[n]=\psi(\Omega+{\alpha-1})\uparrow\uparrow n$$, when $$\alpha$$ is a successor ordinal.

$$\psi(\Omega+\beta)[n]=\psi(\Omega+\beta[n])$$, when $$\beta$$ is a limit ordinal.