## FANDOM

10,835 Pages

Normal ordinals

Catching function
$$\psi(\psi_I(0))$$ $$C(\omega)$$
$$\psi(\psi_I(\psi_I(0)))$$ $$C(\omega2)$$
$$\psi(\psi_I(I))$$ $$C(\omega^2)$$
$$\psi(\psi_I(I^\omega))$$ $$C(\omega^\omega)$$
$$\psi(\psi_I(I^{I\omega}))$$ $$C(\omega^{\omega^2})$$
$$\psi(\psi_I(I^{I^\omega}))$$ $$C(\omega^{\omega^\omega})$$
$$\psi(\psi_I(\varepsilon_{I+1}))$$ $$C(\varepsilon_0)$$
$$\psi(\psi_I(\varepsilon_{I+\Omega_\omega}))$$ $$C(\varepsilon_0+1)$$
$$\psi(\psi_I(\varepsilon_{I2}))$$ $$C(\varepsilon_0\omega)$$
$$\psi(\psi_I(\varepsilon_{I2}\Omega_\omega))$$ $$C(\varepsilon_0\omega+1)$$

## Beyond $$C(\varepsilon_0\omega+1)$$

Thanks to Hyp cos I know the right value of $$C(\varepsilon_0\omega+1)$$, now I can continue my blog post.

Normal ordinals Catching function
$$\psi(\psi_I(\varepsilon_{I2}\Omega_{\Omega_\omega}))$$ $$C(\varepsilon_0\omega+2)$$
$$\psi(\psi_I(\varepsilon_{I2}\psi_I(0)))$$ $$C(\varepsilon_0\omega+\omega)$$
$$\psi(\psi_I(\varepsilon_{I2}I))$$ $$C(\varepsilon_0\omega^2)$$
$$\psi(\psi_I(\varepsilon_{I2}I^\omega))$$ $$C(\varepsilon_0\omega^\omega)$$
$$\psi(\psi_I(\varepsilon_{I2}\varepsilon_{I+1}))$$ $$C(\varepsilon_0^2)$$
$$\psi(\psi_I(\varepsilon_{I2}^\omega))$$ $$C(\varepsilon_0^\omega)$$
$$\psi(\psi_I(\varepsilon_{I2+1}))$$ $$C(\varepsilon_1)$$
$$\psi(\psi_I(\varepsilon_{I2+\Omega_\omega}))$$ $$C(\varepsilon_1+1)$$
$$\psi(\psi_I(\varepsilon_{I3}))$$ $$C(\varepsilon_2)$$
$$\psi(\psi_I(\varepsilon_{I\omega}))$$ $$C(\varepsilon_\omega)$$
$$\psi(\psi_I(\varepsilon_{I^2}))$$ $$C(\varepsilon_\omega\omega)$$
$$\psi(\psi_I(\varepsilon_{I^2+1}))$$ $$C(\varepsilon_{\omega+1})$$
$$\psi(\psi_I(\varepsilon_{\varepsilon_{I+1}}))$$ $$C(\varepsilon_{\varepsilon_0})$$
$$\psi(\psi_I(\zeta_{I+1}))$$ $$C(\zeta_0)$$
$$\psi(\psi_I(\varphi(\omega,{I+1})))$$ $$C(\varphi(\omega,0))$$
$$\psi(\psi_I(\Omega_{I+1}))$$ $$C(\vartheta(\Omega))$$
$$\psi(\psi_I(\Omega_{I+\omega}))$$ $$C(C(0))$$
$$\psi(\psi_I(\Omega_{I2}))$$ $$C(C(0)\omega)$$
$$\psi(\psi_{I_2}(0))$$ $$C(C(\omega))$$
$$\psi(\psi_{I_\omega}(0))$$ $$C(\Omega)$$
$$\psi(\psi_{I_I}(I))$$ $$C(\Omega+\psi_{I_\omega}(0)\omega^2)$$
$$\psi(\psi_{I_I}(I+1))$$ $$C(\Omega+\psi_{I_\omega}(1))$$
$$\psi(\psi_{I_I}(I2))$$ $$C(\Omega+\psi_{I_\omega}(1)\omega)$$
$$\psi(\psi_{I_I}(I\omega))$$ $$C(\Omega+\psi_{I_\omega}(\omega))$$
$$\psi(\psi_{I_I}(I^2))$$ $$C(\Omega+\psi_{I_\omega}(\omega)\omega)$$
$$\psi(\psi_{I_I}(I^I))$$ $$C(\Omega+\psi_{I_\omega}(\omega^\omega)\omega)$$
$$\psi(\psi_{I_I}(\varepsilon_{I+1}))$$ $$C(\Omega+\psi_{I_\omega}(\varepsilon_0))$$
$$\psi(\psi_{I_I}(\Omega_{I+1}))$$ $$C(\Omega+\psi_{I_\omega}(\Omega))$$
$$\psi(\psi_{I_I}(I_2))$$ $$C(\Omega+\psi_{I_\omega}(I))$$
$$\psi(\psi_{I_I}(I_\omega))$$ $$C(\Omega+\psi_{I_\omega}(I_\omega))$$
$$\psi(\psi_{I_I}(I_I))$$ $$C(\Omega+\psi_{I_\omega}(I_\omega)\omega)$$
$$\psi(\psi_{I_{I+1}}(0))$$ $$C(\Omega+\psi_{I_{\omega+1}}(0))$$
$$\psi(\psi_{I_{I+\omega}}(0))$$ $$C(\Omega+\psi_{I_{\omega2}}(0))$$
$$\psi(\psi_{I_{I2}}(0))$$ $$C(\Omega+\psi_{I_{\omega2}}(0)\omega)$$
$$\psi(\psi_{I_{I2+1}}(0))$$ $$C(\Omega+\psi_{I_{\omega2+1}}(0))$$
$$\psi(\psi_{I_{I^2+1}}(0))$$ $$C(\Omega+\psi_{I_{\omega^2+1}}(0))$$
$$\psi(\psi_{I_{\varepsilon_{I+1}}}(0))$$ $$C(\Omega+\psi_{I_{\varepsilon_0}}(0))$$
$$\psi(\psi_{I_{\Omega_{I+1}}}(0))$$ $$C(\Omega+\psi_{I_{\Omega}}(0))$$
$$\psi(\psi_{I_{I_2}}(0))$$ $$C(\Omega+\psi_{I_{I}}(0))$$
$$\psi(\psi_{I_{I_\omega}}(0))$$ $$C(\Omega2)$$
$$\psi(\psi_{I_{I_{I_\omega}}}(0))$$

$$C(\Omega3)$$

$$\psi(\psi_{I(1,0)}(0))$$ $$C(\Omega\omega)$$
$$\psi(\psi_{I(1,0)}(\Omega_\omega))$$ $$C(\Omega\omega+1)$$
$$\psi(\psi_{I(1,0)}(I))$$ $$C(\Omega\omega+\psi_{I(1,0)}(0)\omega)$$
$$\psi(\psi_{I(1,0)}(I(1,0)))$$ $$C(\Omega(\omega+1))$$
$$\psi(\psi_{I(1,0)}(\varepsilon_{I(1,0)+1}))$$ $$C(\Omega(\varepsilon_0))$$
$$\psi(\psi_{I(1,\omega)}(0))$$ $$C(\Omega^2)$$
$$\psi(\psi_{I(2,0)}(0))$$ $$C(\Omega^2\omega)$$
$$\psi(\psi_{I(3,0)}(0))$$ $$C(\Omega^3\omega)$$
$$\psi(\psi_{I(\omega,0)}(0))$$ $$C(\Omega^\omega)$$

## Beyond linear arrays

Normal ordinals Catching function
$$\psi(\psi_{I(\omega,0)}(0))$$ $$C(\Omega^\omega)$$
$$\psi(\psi_{I(1,0,0)}(0))$$ $$C(\Omega^\omega\omega)$$
$$\psi(\psi_{I(1,0,\omega)}(0))$$ $$C(\Omega^{\omega+1})$$
$$\psi(\psi_{I(1,1,\omega)}(0))$$ $$C(\Omega^{\omega+2})$$
$$\psi(\psi_{I(1,\omega,0)}(0))$$ $$C(\Omega^{\omega2})$$
$$\psi(\psi_{I(2,0,0)}(0))$$ $$C(\Omega^{\omega2}\omega)$$
$$\psi(\psi_{I(\omega,0,0)}(0))$$ $$C(\Omega^{\omega^2})$$
$$\psi(\psi_{I(1,0,0,0)}(0))$$ $$C(\Omega^{\omega^2}\omega)$$
$$\psi(\psi_{\chi(M^\omega)}(0))$$ $$C(\Omega^{\omega^\omega})$$
$$\psi(\psi_{\chi(M^M)}(0))$$ $$C(\Omega^{\omega^\omega}\omega)$$
$$\psi(\psi_{\chi(\varepsilon_{M+1})}(0))$$ $$C(\Omega^{\varepsilon_0})$$
$$\psi(\psi_{\chi(\Omega_{M+1})}(0))$$ $$C(\Omega^{\Gamma_0})$$
$$\psi(\psi_{\chi(\Omega_{M+\omega})}(0))$$

$$C(\Omega^{C(0)})$$

$$\psi(\psi_{\chi(I_{M+1})}(0))$$ $$C(\Omega^{C(\omega)})$$
$$\psi(\psi_{\chi(M_2)}(0))$$ $$C(\Omega^{C(\Omega^\omega\omega)})$$
$$\psi(\psi_{\chi(M_3)}(0))$$ $$C(\Omega^{C(\Omega^{C(\Omega^\omega\omega)})})$$
$$\psi(\psi_{\chi(M_\omega)}(0))$$ $$C(\Omega^\Omega)$$

## Beyond!

Normal ordinals Catching function
$$\psi(\psi_{\chi(M_\omega)}(0))$$ $$C(\Omega^\Omega)$$
$$\psi(\psi_{\chi(M_{M_\omega})}(0))$$ $$C(\Omega^{\Omega}2)$$
$$\psi(\psi_{\chi(M(1,0))}(0))$$ $$C(\Omega^{\Omega}\omega)$$
$$\psi(\psi_{\chi(M(\omega,0))}(0))$$ $$C(\Omega^{\Omega\omega})$$
$$\psi(\psi_{\chi(M(\Xi(3,0),0))}(0))$$ $$C(\Omega^{\Omega\omega}\omega)$$
$$\psi(\psi_{\chi(M(\Xi(3,\omega),0))}(0))$$ $$C(\Omega^{\Omega^2})$$
$$\psi(\Psi_{\Xi(\omega,0)}(0))$$ $$C(\Omega^{\Omega^\omega})$$
$$\psi(\Psi_{\Xi(\Omega_\omega,0)}(0))$$ $$C(\Omega^{\Omega^\omega}+1)$$
$$\psi(\Psi_{\Xi(\psi_{I_\omega}(0),0)}(0))$$ $$C(\Omega^{\Omega^\omega}+\Omega)$$
$$\psi(\Psi_{\Xi(\Xi(\omega,0),0)}(0))$$ $$C(\Omega^{\Omega^\omega}\omega)$$

$$\psi(\Psi_{\Xi(K,0)}(0))$$

$$C(\Omega^{\Omega^\omega}\omega^2)$$
$$\psi(\Psi_{\Xi(\varepsilon_{K+1},0)}(0))$$ $$C(\Omega^{\Omega^\omega}\varepsilon_0)$$
$$\psi(\Psi_{\Xi(K_\omega,0)}(0))$$ $$C(\Omega^{\Omega^\omega+1})$$

The limit of NAN in Dollar function and the limit of PNAN (R function) might be $$C(\varepsilon_{\Omega+1})$$.