User:Hyp cos/Catching Function Analysis

First, why do I write a "user page" instead of "user blog"? Because It's easier to edit (section edit) for user page. Imagine this: a user page can contain some sections, and they contain some subsections each,... and I can edit all of them, and each subsubsubsubsection can be as long as a whole user blog. It must be a long time to wait for the LaTeX loading :( =Catching Function= I have already shown things up to \(C(\varepsilon_0\omega)\) in this blog.

From \(C(\varepsilon_0\omega)\) to \(C(\varepsilon_1)\)
Since catching function is defined as SGH-catching-FGH-points, I should compare SGH with FGH too. I use [k] for the k-th ordinal in fundamental sequence. Something may not have fundamental sequence, such as uncountable cardinals, but since cardinals in ordinal collapsing functions mean the limit of finity nests, I define \(\alpha[k]\) means k nests if \(\alpha\) is an uncountable cardinal.

How to get \(C(\varepsilon_0\omega+1)\)
\begin{eqnarray*} \text{FGH} & & \text{SGH ordinals} \\ f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}) \\ 2f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2})2 \\ f_2^nf_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+1) \\ f_3f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi(\varepsilon_{I2})) \\ f_3^nf_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega) \\ f_4^nf_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^2) \\ f_nf_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^\omega) \\ f_\omega f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\psi(\varepsilon_{I2})}) \\ f_\omega^nf_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^\Omega) \\ f^n_{\omega+1}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\Omega+1}) \\ f^n_{\omega2}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\Omega2}) \\ f_{\omega n}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\Omega\omega}) \\ f^n_{\omega^2}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\Omega^2}) \\ f^n_{\omega^\omega}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega^{\Omega^\Omega}) \\ f_{\varepsilon_0[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{\Omega+1}) \\ f_{\varepsilon_0}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{\Omega+\psi(\varepsilon_{I2})}) \\ f^n_{\varepsilon_0}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{\Omega2}) \\ f_{\zeta_0[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\zeta_{\Omega+1}) \\ f_{\Gamma_0[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\vartheta_1(\Omega_2)) \\ f_{\vartheta(\Omega_2)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\vartheta_1(\Omega_3)) \\ f_{\vartheta(\Omega_n)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\vartheta_1(\Omega_\omega)) \\ f_{\psi(\psi_I(0))[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\psi_I(0))) \\ f_{\psi(\psi_I(0))[2n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\psi_I(1))) \\ f^n_{\psi(\psi_I(0))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\psi_I(\Omega))) \\ f^n_{\psi(\psi_I(\Omega))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\psi_I(\Omega_2))) \\ f_{\psi(\psi_I(\Omega_n))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\psi_I(\Omega_\omega))) \\ f_{\psi(I)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I)) \\ f_{\psi(I)[2n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I2)) \\ f^n_{\psi(I)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I\Omega)) \\ f_{\psi(I\Omega_n)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I\Omega_\omega)) \\ f_{\psi(I^2)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I^2)) \\ f_{\psi(I^n)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(I^\omega)) \\ f_{\psi(\varepsilon_{I+1})[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I+1})) \\ f^n_{\psi(\varepsilon_{I+1})}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I+\Omega})) \\ f_{\psi(\varepsilon_{I2})[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})) \end{eqnarray*}

But those above just warm up your mind.

\begin{eqnarray*} \text{FGH} & & \text{SGH ordinals} \\ f^n_2f_{\psi(\varepsilon_{I2})[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})+1) \\ f^2_{\psi(\varepsilon_{I2})[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})2) \\ f^n_{\psi(\varepsilon_{I2})[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})\omega) \\ f^n_{\psi(\varepsilon_{I2})[n]+1}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})\Omega) \\ f_{\psi(\varepsilon_{I2})[n]2}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2})^2) \\ f_{\psi(\varepsilon_{I2}[n]+1)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2}+1)) \\ f^n_{\psi(\varepsilon_{I2}[n]+\omega)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2}+\Omega)) \\ f^n_{\psi(\varepsilon_{I2}[n]+\psi(\varepsilon_{I2}[n]+\omega))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_1(\varepsilon_{I2}+\psi_1(\varepsilon_{I2}+\Omega))) \\ f_{\psi(\varepsilon_{I2}[n]+\Omega)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega_2) \\ f_{\psi(\varepsilon_{I2}[n]+\Omega_n)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\Omega_\omega) \\ f_{\psi(\varepsilon_{I2}[n]+\psi_I(0)[n])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(0)) \\ f_{\psi(\varepsilon_{I2}[n]+\psi_I(I)[n])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(I)) \\ f_{\psi(\varepsilon_{I2}[n]+\psi_I(\varepsilon_{I+1}[n]))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(\varepsilon_{I+1})) \\ f^n_{\psi(\varepsilon_{I2}[n]+\psi_I(\varepsilon_{I+1}))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(\varepsilon_{I+\Omega})) \\ f_{\psi(\varepsilon_{I2}[n]+\psi_I(\varepsilon_{I2})[n])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(\varepsilon_{I2})) \\ f_{\psi(\varepsilon_{I2}[n]+\psi_I(\varepsilon_{I2}[n]+\psi_I(\varepsilon_{I2}[n])))}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\psi_I(\varepsilon_{I2}+\psi_I(\varepsilon_{I2}))) \\ f_{\psi(\varepsilon_{I2}[n]+I)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+I) \\ f_{\psi(\varepsilon_{I2}[n]+I)[2n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+I2) \\ f_{\psi(\varepsilon_{I2}[n]+I\Omega_n)}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+I\Omega_\omega) \\ f_{\psi(\varepsilon_{I2}[n]+I^2)[n]}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+I^2) \\ f^n_{\psi(\varepsilon_{I2}[n]+\varepsilon_{I+1})}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{I+\Omega}) \end{eqnarray*}

Things above can be skip if we compare SGH to FGH from "inside". Pay attention to these below. And notice that \(\psi_I(\varepsilon_{I2}[k]+\varepsilon_{I+\psi_I(\varepsilon_{I2}[k])})=\psi_I(\varepsilon_{I+\psi_I(\varepsilon_{I2}[k])})=\psi_I(\varepsilon_{I2}[k+1])\).

\begin{eqnarray*} \text{FGH} & & \text{SGH ordinals} \\ f_{\psi(\varepsilon_{I+\psi_I(\varepsilon_{I2}[n])})}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{I+\psi_I(\varepsilon_{I2})}) \\ f_{\psi(\varepsilon_{I2}[n+2])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}+\varepsilon_{I+\psi_I(\varepsilon_{I2}+\varepsilon_{I+\psi_I(\varepsilon_{I2})})}) \\ f_{\psi(\varepsilon_{I2}[2n])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}2) \\ f_{\psi(\varepsilon_{I2}[2n+1])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}2+\varepsilon_{I+\psi_I(\varepsilon_{I2}2)}) \\ f_{\psi(\varepsilon_{I2}[3n])}f_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}3) \\ f^2_{\psi(\varepsilon_{I2})}(n) & & \psi(\varepsilon_{I2}\psi(\varepsilon_{I2})) \\ f_{\psi(\varepsilon_{I2})+1}(n) & & \psi(\varepsilon_{I2}\Omega) \\  f_{\psi(\varepsilon_{I2})+\omega+1}(n) & & \psi(\varepsilon_{I2}\Omega+\Omega^\Omega) \\ f_{\psi(\varepsilon_{I2})2}(n) & & \psi(\varepsilon_{I2}\Omega+\psi_1(\varepsilon_{I2})) \\ f_{\psi(\varepsilon_{I2}+1)}(n) & & \psi(\varepsilon_{I2}\Omega+\psi_1(\varepsilon_{I2}+1)) \\ f_{\psi(\varepsilon_{I2}+\Omega)}(n) & & \psi(\varepsilon_{I2}\Omega+\Omega_2) \\ f_{\psi(\varepsilon_{I2}+\Omega_\omega)}(n) & & \psi(\varepsilon_{I2}\Omega+\Omega_\omega) \\ f_{\psi(\varepsilon_{I2}+I)}(n) & & \psi(\varepsilon_{I2}\Omega+I) \\ f_{\psi(\varepsilon_{I2}2)}(n) & & \psi(\varepsilon_{I2}\Omega+\varepsilon_{I2}) \\ f_{\psi(\varepsilon_{I2}\omega)}(n) & & \psi(\varepsilon_{I2}\Omega\omega) \\ f_{\psi(\varepsilon_{I2}\Omega)}(n) & & \psi(\varepsilon_{I2}\Omega_2) \\ f_{\psi(\varepsilon_{I2}\Omega_\omega)}(n) & & \psi(\varepsilon_{I2}\Omega_\omega) \end{eqnarray*}

So we get \(C(\varepsilon_0\omega+1)\) now.

Up to \(C(\varepsilon_0^2)\)
\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\varepsilon_0\omega) &=& \psi(\varepsilon_{I2}) \\ C(\varepsilon_0\omega+1) &=& \psi(\varepsilon_{I2}\Omega_\omega) \\ C(\varepsilon_0\omega+\omega) &=& \psi(\varepsilon_{I2}\psi_I(0)) \\ C(\varepsilon_0\omega2) &=& \psi(\varepsilon_{I2}\psi_I(\varepsilon_{I2})) \\ C(\varepsilon_0\omega2+1) &=& \psi(\varepsilon_{I2}\psi_I(\varepsilon_{I2}\Omega_\omega)) \\ C(\varepsilon_0\omega3) &=& \psi(\varepsilon_{I2}\psi_I(\varepsilon_{I2}\psi_I(\varepsilon_{I2}))) \\ C(\varepsilon_0\omega^2) &=& \psi(\varepsilon_{I2}I) \\ C(\varepsilon_0\omega^2+1) &=& \psi(\varepsilon_{I2}I\Omega_\omega) \\ C(\varepsilon_0\omega^3) &=& \psi(\varepsilon_{I2}I^2) \\ C(\varepsilon_0\omega^\omega) &=& \psi(\varepsilon_{I2}I^\omega) \\ C(\varepsilon_0\omega^\omega+1) &=& \psi(\varepsilon_{I2}I^{\Omega_\omega}) \\ C(\varepsilon_0\omega^\omega2) &=& \psi(\varepsilon_{I2}I^{\psi_I(\varepsilon_{I2}I^\omega)}) \\ C(\varepsilon_0\omega^{\omega+1}) &=& \psi(\varepsilon_{I2}I^I) \\ C(\varepsilon_0\omega^{\omega2}) &=& \psi(\varepsilon_{I2}I^{I+\omega}) \\ C(\varepsilon_0\omega^{\omega2+1}) &=& \psi(\varepsilon_{I2}I^{I2}) \\ C(\varepsilon_0\omega^{\omega^2}) &=& \psi(\varepsilon_{I2}I^{I\omega}) \\ C(\varepsilon_0\omega^{\omega^2+1}) &=& \psi(\varepsilon_{I2}I^{I^2}) \\ C(\varepsilon_0^2) &=& \psi(\varepsilon_{I2}\varepsilon_{I+1}) \end{eqnarray*}

Get \(C(\varepsilon_0^2+1)\)
\begin{eqnarray*} \text{FGH} & & \text{SGH Ordinal} \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}) \\ f_2^nf_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+1) \\ f^n_{\omega}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\Omega^\Omega) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n])}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\psi_1(\varepsilon_{I2}\varepsilon_{I+1})) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+\Omega)}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\Omega_2) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+\psi_I(0)[n])}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\psi_I(0)) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+\psi_I(\varepsilon_{I2}\varepsilon_{I+1}[n]))}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\psi_I(\varepsilon_{I2}\varepsilon_{I+1})) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+I[n])}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+I) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+I)}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+I\psi(\varepsilon_{I2}\varepsilon_{I+1})) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+I\Omega_n)}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+I\Omega_\omega) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]+\varepsilon_{I+1}[n])}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}+\varepsilon_{I+1}) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]2)}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}2) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]^2)}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}^2) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[n]^{\varepsilon_{I+1}[n]})}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+1}^{\varepsilon_{I+1}}) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1}[2n])}f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+2}) \\ f^2_{\psi(\varepsilon_{I2}\varepsilon_{I+1})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+\psi(\varepsilon_{I2}\varepsilon_{I+1})}) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+1})+1}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+\Omega}) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+\Omega})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+\Omega_2}) \\ f_{\psi(\varepsilon_{I2}\varepsilon_{I+\Omega_\omega})}(n) & & \psi(\varepsilon_{I2}\varepsilon_{I+\Omega_\omega}) \end{eqnarray*}

Continue to \(C(\varepsilon_0^\omega)\)
So we get

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\varepsilon_0^2+1) &=& \psi(\varepsilon_{I2}\varepsilon_{I+\Omega_\omega}) \\ C(\varepsilon_0^2+\omega) &=& \psi(\varepsilon_{I2}\varepsilon_{I+\Omega_{\Omega_\omega}}) \\ C(\varepsilon_0^22) &=& \psi(\varepsilon_{I2}\varepsilon_{I+\psi_I(\varepsilon_{I2}\varepsilon_{I+1}})) \\ C(\varepsilon_0^2\omega) &=& \psi(\varepsilon_{I2}^2) \\ C(\varepsilon_0^2+1) &=& \psi(\varepsilon_{I2}^2\Omega_\omega) \\ C(\varepsilon_0^2\omega) &=& \psi(\varepsilon_{I2}^2I) \\ C(\varepsilon_0^2\omega^\omega) &=& \psi(\varepsilon_{I2}^2I^\omega) \\ C(\varepsilon_0^2\omega^{\omega+1}) &=& \psi(\varepsilon_{I2}^2I^I) \\ C(\varepsilon_0^3) &=& \psi(\varepsilon_{I2}^2\varepsilon_{I+1}) \\ C(\varepsilon_0^3+1) &=& \psi(\varepsilon_{I2}^2\varepsilon_{I+\Omega_\omega}) \\ C(\varepsilon_0^3\omega) &=& \psi(\varepsilon_{I2}^3) \\ C(\varepsilon_0^4) &=& \psi(\varepsilon_{I2}^3\varepsilon_{I+1}) \\ C(\varepsilon_0^\omega) &=& \psi(\varepsilon_{I2}^\omega) \end{eqnarray*}

Up to \(C(\varepsilon_1)\)
\(C(\varepsilon_0^\omega+1)=\psi(\varepsilon_{I2}^{\Omega_\omega})\), and we go along with the way we get \(\varepsilon_0\). Everything goes the same.

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\varepsilon_0^\omega+1) &=& \psi(\varepsilon_{I2}^{\Omega_\omega}) \\ C(\varepsilon_0^\omega2) &=& \psi(\varepsilon_{I2}^{\psi_I(\varepsilon_{I2}^\omega)}) \\ C(\varepsilon_0^\omega\omega) &=& \psi(\varepsilon_{I2}^I) \\ C(\varepsilon_0^\omega\omega^2) &=& \psi(\varepsilon_{I2}^II) \\ C(\varepsilon_0^\omega\omega^\omega) &=& \psi(\varepsilon_{I2}^II^I) \\ C(\varepsilon_0^{\omega+1}) &=& \psi(\varepsilon_{I2}^I\varepsilon_{I+1}) \\ C(\varepsilon_0^{\omega+1}\omega) &=& \psi(\varepsilon_{I2}^{I+1}) \\ C(\varepsilon_0^{\omega+2}\omega) &=& \psi(\varepsilon_{I2}^{I+2}) \\ C(\varepsilon_0^{\omega2}\omega) &=& \psi(\varepsilon_{I2}^{I2}) \\ C(\varepsilon_0^{\omega^2}\omega) &=& \psi(\varepsilon_{I2}^{I^2}) \\ C(\varepsilon_0^{\omega^\omega}\omega) &=& \psi(\varepsilon_{I2}^{I^I}) \\ C(\varepsilon_0^{\varepsilon_0}) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I+1}}) \\ C(\varepsilon_0^{\varepsilon_0}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}}) \\ C(\varepsilon_0^{\varepsilon_0+1}) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}}\varepsilon_{I+1}) \\ C(\varepsilon_0^{\varepsilon_0+1}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}+1}) \\ C(\varepsilon_0^{\varepsilon_02}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}2}) \\ C(\varepsilon_0^{\varepsilon_0^2}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}^2}) \\ C(\varepsilon_0^{\varepsilon_0^{\omega}}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}^I}) \\ C(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}\omega) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}^{\varepsilon_{I2}}}) \\ C(\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}) &=& \psi(\varepsilon_{I2}^{\varepsilon_{I2}^{\varepsilon_{I2}^{\varepsilon_{I+1}}}}) \\ C(\varepsilon_1) &=& \psi(\varepsilon_{I2+1}) \end{eqnarray*}

It seems a bit strange - when it grows from \(C(\varepsilon_0)\) to \(C(\varepsilon_0\omega)\), the normal notation grows from \(\psi(\varepsilon_{I+1})\) to \(\psi(\varepsilon_{I2})\), this is a big step; but when it grows from \(C(\varepsilon_0\omega)\) to \(C(\varepsilon_1)\), the normal notation grows from \(\psi(\varepsilon_{I2})\) to only \(\psi(\varepsilon_{I2+1})\) - it seems a small step.

From \(C(\varepsilon_1)\) to C(BHO)
When the preparations are enough, we can go rapidly now.

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\varepsilon_1+1) &=& \psi(\varepsilon_{I2+\Omega_\omega}) \\ C(\varepsilon_12) &=& \psi(\varepsilon_{I2+\psi_I(\varepsilon_{I2+1})}) \\ C(\varepsilon_1\omega) &=& \psi(\varepsilon_{I3}) \\ C(\varepsilon_1^2) &=& \psi(\varepsilon_{I3}\varepsilon_{I2+1}) \\ C(\varepsilon_1^\omega) &=& \psi(\varepsilon_{I3}^\omega) \\ C(\varepsilon_1^{\varepsilon_1}\omega) &=& \psi(\varepsilon_{I3}^{\varepsilon_{I3}}) \\ C(\varepsilon_2) &=& \psi(\varepsilon_{I3+1}) \\ C(\varepsilon_2\omega) &=& \psi(\varepsilon_{I4}) \\ C(\varepsilon_3\omega) &=& \psi(\varepsilon_{I5}) \\ C(\varepsilon_\omega) &=& \psi(\varepsilon_{I\omega}) \\ C(\varepsilon_\omega\omega) &=& \psi(\varepsilon_{I^2}) \\ C(\varepsilon_{\omega+1}) &=& \psi(\varepsilon_{I^2+1}) \\ C(\varepsilon_{\omega+1}\omega) &=& \psi(\varepsilon_{I^2+I}) \\ C(\varepsilon_{\omega+2}\omega) &=& \psi(\varepsilon_{I^2+I2}) \\ C(\varepsilon_{\omega2}) &=& \psi(\varepsilon_{I^2+I\omega}) \\ C(\varepsilon_{\omega2}\omega) &=& \psi(\varepsilon_{I^22}) \\ C(\varepsilon_{\omega3}\omega) &=& \psi(\varepsilon_{I^23}) \\ C(\varepsilon_{\omega^2}\omega) &=& \psi(\varepsilon_{I^3}) \\ C(\varepsilon_{\omega^3}\omega) &=& \psi(\varepsilon_{I^4}) \\ C(\varepsilon_{\omega^\omega}) &=& \psi(\varepsilon_{I^\omega}) \\ C(\varepsilon_{\omega^\omega}\omega) &=& \psi(\varepsilon_{I^I}) \\ C(\varepsilon_{\omega^{\omega^\omega}}\omega) &=& \psi(\varepsilon_{I^{I^I}}) \\ C(\varepsilon_{\varepsilon_0}) &=& \psi(\varepsilon_{\varepsilon_{I+1}}) \\ C(\varepsilon_{\varepsilon_0}\omega) &=& \psi(\varepsilon_{\varepsilon_{I2}}) \\ C(\varepsilon_{\varepsilon_1}\omega) &=& \psi(\varepsilon_{\varepsilon_{I3}}) \\ C(\varepsilon_{\varepsilon_{\varepsilon_0}}\omega) &=& \psi(\varepsilon_{\varepsilon_{\varepsilon_{I2}}}) \\ C(\zeta_0) &=& \psi(\zeta_{I+1}) \end{eqnarray*}

Now we start using \(\psi_{\Omega_{I+1}}\) function. \(\psi_{\Omega_{I+1}}(0)=I\uparrow\uparrow\omega\) and \(\psi_{\Omega_{I+1}}(\alpha+1)=\psi_{\Omega_{I+1}}(\alpha)\uparrow\uparrow\omega\). The \(\Omega_{I+1}\) is used as the diagonalizer. Similarly \(\psi_{\Omega_{\pi+1}}(0)=\Omega_\pi\uparrow\uparrow\omega\) and \(\psi_{\Omega_{\pi+1}}(\alpha+1)=\psi_{\Omega_{\pi+1}}(\alpha)\uparrow\uparrow\omega\). The \(\Omega_{\pi+1}\) is used as the diagonalizer. Using these notation, I get \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\zeta_0) &=& \psi(\Omega_{I+1}) \\ C(\zeta_0+1) &=& \psi(\Omega_{I+1}\Omega_\omega)=\psi(\zeta_{I+\Omega_\omega}) \\ C(\zeta_0\omega) &=& \psi(\Omega_{I+1}I)=\psi(\zeta_{I2}) \\ C(\zeta_0\omega^2) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)I)=\psi(\zeta_{I2}I) \\ C(\zeta_0\omega^\omega) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)I^\omega)=\psi(\zeta_{I2}I^\omega) \\ C(\zeta_0\varepsilon_0) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)\psi_{\Omega_{I+1}}(0)) \\ C(\zeta_0^2) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)\psi_{\Omega_{I+1}}(0)) \\ C(\zeta_0^3) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)^2\psi_{\Omega_{I+1}}(0)) \\ C(\zeta_0^{\zeta_0}) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I)^{\psi_{\Omega_{I+1}}(\Omega_{I+1}I)}) \\ C(\varepsilon_{\zeta_0+1}) &=& \psi(\Omega_{I+1}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}I+1)) \\ C(\zeta_1) &=& \psi(\Omega_{I+1}I+\Omega_{I+1}) \\ C(\zeta_2) &=& \psi(\Omega_{I+1}I2+\Omega_{I+1}) \\ C(\zeta_{\zeta_0}) &=& \psi(\Omega_{I+1}\psi_{\Omega_{I+1}}(\Omega_{I+1})) \\ C(\varphi(3,0)) &=& \psi(\Omega_{I+1}^2) \\ C(\varphi(4,0)) &=& \psi(\Omega_{I+1}^3) \\ C(\varphi(\omega,0)\omega) &=& \psi(\Omega_{I+1}^I) \\ C(\Gamma_0) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}) \\ C(\Gamma_0\omega) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I) \\ C(\varepsilon_{\Gamma_0+1}) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I+\psi_{\Omega_{I+1}}(\Omega_{I+1}^{\Omega_{I+1}}I+1)) \\ C(\varphi(2,\Gamma_0+1)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I+\Omega_{I+1}) \\ C(\varphi(3,\Gamma_0+1)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I+\Omega_{I+1}^2) \\ C(\varphi(\omega,\Gamma_0+1)\omega) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}\psi_{\Omega_{I+1}}(\Omega_{I+1}^{\Omega_{I+1}}+\Omega_{I+1}^I)) \\ C(\varphi(\Gamma_0,1)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I+\Omega_{I+1}^{\psi_{\Omega_{I+1}}(\Omega_{I+1}^{\Omega_{I+1}})}) \\ C(\Gamma_1) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}}I+\Omega_{I+1}^{\Omega_{I+1}}) \\ C(\varphi(1,1,0)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}+1}) \\ C(\varphi(1,\omega,0)\omega) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}+I}) \\ C(\varphi(2,0,0)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}2}) \\ C(\varphi(1,0,0,0)) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}^2}) \\ C(\psi(\Omega^{\Omega^\omega})\omega) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}^I}) \\ C(\psi(\Omega^{\Omega^\Omega})) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}^{\Omega_{I+1}}}) \\ C(\psi(\Omega^{\Omega^{\Omega^\Omega}})) &=& \psi(\Omega_{I+1}^{\Omega_{I+1}^{\Omega_{I+1}^{\Omega_{I+1}}}}) \\ C(\psi(\varepsilon_{\Omega+1})) &=& \psi(\varepsilon_{\Omega_{I+1}+1})=\psi(\psi_{\Omega_{I+2}}(0)) \end{eqnarray*}

From C(BHO) to \(C(C(0)\omega)\)
Since \(C(0)=\psi(\Omega_\omega)\), we know simply that \(C(C(0))=C(\psi(\Omega_\omega))\). Actually, the \(\omega\) in Catching function acts an I in normal notation, the \(\Omega\) in \(C(\psi)\) function acts an \(\Omega_{I+1}\) in normal notation, and the \(\Omega_2\) in \(C(\psi)\) function acts an \(\Omega_{I+2}\) in normal notation. These will lead to \(\Omega_{I+\omega}\). More detailed, \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\varepsilon_{\Omega+1})\omega) &=& \psi(\varepsilon_{\Omega_{I+1}+I})=\psi(\psi_{\Omega_{I+2}}(I)) \\ C(\psi(\varepsilon_{\Omega+1})\omega+1) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I))\Omega_\omega) \\ C(\psi(\varepsilon_{\Omega+1})\omega^2) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I))I) \\ C(\psi(\varepsilon_{\Omega+1})^2) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I))\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(0))) \\ C(\psi(\varepsilon_{\Omega+1})^{\psi(\varepsilon_{\Omega+1})}\omega) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I))^{\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I))}) \\ C(\psi(\varepsilon_{\Omega+1}+1)) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I)+1)) \\ C(\psi(\varepsilon_{\Omega+1}+\Omega)) &=& \psi(\psi_{\Omega_{I+2}}(I)+\Omega_{I+1}) \\ C(\psi(\varepsilon_{\Omega+1}2)) &=& \psi(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+2}}(0)) \\ C(\psi(\varepsilon_{\Omega+1}\omega)) &=& \psi(\psi_{\Omega_{I+2}}(I)\omega) \\ C(\psi(\varepsilon_{\Omega+1}\Omega)) &=& \psi(\psi_{\Omega_{I+2}}(I)\Omega_{I+1}) \\ C(\psi(\varepsilon_{\Omega+1}\Omega^{\Omega})) &=& \psi(\psi_{\Omega_{I+2}}(I)\Omega_{I+1}^{\Omega_{I+1}}) \\ C(\psi(\varepsilon_{\Omega+1}^2)) &=& \psi(\psi_{\Omega_{I+2}}(I)\psi_{\Omega_{I+2}}(0)) \\ C(\psi(\varepsilon_{\Omega+1}^2)\omega) &=& \psi(\psi_{\Omega_{I+2}}(I)^2) \\ C(\psi(\varepsilon_{\Omega+1}^\omega)\omega) &=& \psi(\psi_{\Omega_{I+2}}(I)^I) \\ C(\psi(\varepsilon_{\Omega+1}^\Omega)) &=& \psi(\psi_{\Omega_{I+2}}(I)^{\Omega_{I+1}}) \\ C(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})) &=& \psi(\psi_{\Omega_{I+2}}(I)^{\psi_{\Omega_{I+2}}(0)}) \\ C(\psi(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\omega) &=& \psi(\psi_{\Omega_{I+2}}(I)^{\psi_{\Omega_{I+2}}(I)}) \\ C(\psi(\varepsilon_{\Omega+2})) &=& \psi(\psi_{\Omega_{I+2}}(I+1)) \\ C(\psi(\varepsilon_{\Omega+2})\omega) &=& \psi(\psi_{\Omega_{I+2}}(I2)) \\ C(\psi(\varepsilon_{\Omega+3})) &=& \psi(\psi_{\Omega_{I+2}}(I2+1)) \\ C(\psi(\varepsilon_{\Omega+\omega})) &=& \psi(\psi_{\Omega_{I+2}}(I\omega)) \\ C(\psi(\varepsilon_{\Omega2})) &=& \psi(\psi_{\Omega_{I+2}}(\Omega_{I+1})) \\ C(\psi(\varepsilon_{\Omega^\Omega})) &=& \psi(\psi_{\Omega_{I+2}}(\Omega_{I+1}^{\Omega_{I+1}})) \\ C(\psi(\varepsilon_{\varepsilon_{\Omega+1}})) &=& \psi(\psi_{\Omega_{I+2}}(\psi_{\Omega_{I+2}}(0))) \\ C(\psi(\varepsilon_{\varepsilon_{\Omega2}})) &=& \psi(\psi_{\Omega_{I+2}}(\psi_{\Omega_{I+2}}(\Omega_{I+1}))) \\ C(\psi(\Omega_2))=C(\psi(\zeta_{\Omega+1})) &=& \psi(\Omega_{I+2}) \\ C(\psi(\Omega_2^{\Omega_2})) &=& \psi(\Omega_{I+2}^{\Omega_{I+2}}) \\ C(\psi(\varepsilon_{\Omega_2+1})) &=& \psi(\psi_{\Omega_{I+3}}(0)) \\ C(\psi(\Omega_3)) &=& \psi(\Omega_{I+3}) \\ C(\psi(\Omega_4)) &=& \psi(\Omega_{I+4}) \\ C(C(0))=C(\psi(\Omega_\omega)) &=& \psi(\Omega_{I+\omega}) \\ C(\psi(\Omega_\omega)+1) &=& \psi(\Omega_{I+\Omega_\omega}) \\ C(\psi(\Omega_\omega)2) &=& \psi(\Omega_{I+\psi_I(\Omega_{I+\omega})}) \\ C(\psi(\Omega_\omega)\omega) &=& \psi(\Omega_{I2}) \end{eqnarray*}

From \(C(C(0)\omega)\) to \(C(C(\omega))\)
First thing in this part is to get \(C(\psi(\Omega_\omega)\omega+1)\). So let's compare FGH with SGH. We use \(\psi(\Omega_{I2}[k])\) as \(\psi(\Omega_{I2}[0])=\psi(I)\) and \(\psi(\Omega_{I2}[k+1])=\psi(\Omega_{I+\psi_I(\Omega_{I2}[k])})\).

\begin{eqnarray*} \text{FGH} & & \text{SGH Ordinal} \\ f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}) \\ f^n_2f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+1) \\ f^n_\omega f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega^\Omega) \\ f_{\psi(\Omega)[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_2)) \\ f_{\psi(\Omega_{I2})[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})) \\ f^n_{\omega}f_{\psi(\Omega_{I2})[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})+\Omega^{\Omega}) \\ f^2_{\psi(\Omega_{I2})[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})2) \\ f^n_{\psi(\Omega_{I2})[n]+1}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})\Omega) \\ f_{\psi(\Omega_{I2})[n]2}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})^2) \\ f_{\psi(\Omega_{I2})[n]^2}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})^{\psi_1(\Omega_{I2})}) \\ f_{\psi(\Omega_{I2})[n]^{\psi(\Omega_{I2})[n]}}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2})^{\psi_1(\Omega_{I2})^{\psi_1(\Omega_{I2})}}) \\ f_{\psi(\Omega_{I2}[n]+1)[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2}+1)) \\ f^n_{\psi(\Omega_{I2}[n]+1)}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2}+\Omega)) \\ f_{\psi(\Omega_{I2}[n]+\psi(\Omega_{I2}[n]))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_1(\Omega_{I2}+\psi_1(\Omega_{I2}))) \\ f_{\psi(\Omega_{I2}[n]+\Omega)[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_2) \\ f_{\psi(\Omega_{I2}[n]+\Omega_n)}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_\omega) \\ f_{\psi(\Omega_{I2}[n]+\psi_I(\Omega_{I2}[n]))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_I(\Omega_{I2})) \\ f_{\psi(\Omega_{I2}[n]+I)[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+I) \\ f_{\psi(\Omega_{I2}[n]+\varepsilon_{I+1})[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\varepsilon_{I+1}) \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega})) \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(0)[n]}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(0)})) \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(I)[n]}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(I)})) \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+1})[n]}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+1})})) \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+n})}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+\omega})})) \\ f_{\psi(\Omega_{I2}[n+1])}f_{\psi(\Omega_{I2})}(n)= & & \\ f_{\psi(\Omega_{I2}[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})[n]}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})})) \\ f_{\psi(\Omega_{I2}[n+2])}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})}))})) \\ f_{\psi(\Omega_{I2}[2n])}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})) \\ f_{\psi(\Omega_{I2}[2n]+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+n})) \\ f_{\psi(\Omega_{I2}[2n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+n})}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+\omega})})) \\ f_{\psi(\Omega_{I2}[2n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}[n])}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})})) \\ f_{\psi(\Omega_{I2}[2n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}[n+1])}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})}))})) \\ f_{\psi(\Omega_{I2}[2n+1])}f_{\psi(\Omega_{I2})}(n)= & & \\ f_{\psi(\Omega_{I2}[2n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}[2n])}))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}))})) \\ f_{\psi(\Omega_{I2}[2n+2])}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}) \\ & & +\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}))}))})) \\ f_{\psi(\Omega_{I2}[3n])}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})2) \\ f_{\psi(\Omega_{I2}[n^2])}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\omega) \\ f^2_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\psi(\Omega_{I2})) \\ f_{\psi(\Omega_{I2})+1}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\Omega) \\ f_{\psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\Omega_\omega)}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\Omega_\omega) \end{eqnarray*}

Then...

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\Omega_\omega)\omega+1) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\Omega_\omega) \\ C(\psi(\Omega_\omega)\omega+\omega^2) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\psi_I(I)) \\ C(\psi(\Omega_\omega)\omega^2) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})I) \\ C(\psi(\Omega_\omega)\omega^\omega) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})I^\omega) \\ C(\psi(\Omega_\omega)^2) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})\psi_{\Omega_{I+1}}(\Omega_{I+\omega})) \\ C(\psi(\Omega_\omega)^2\omega) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})^2) \\ C(\psi(\Omega_\omega+1)) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}+1)) \\ C(\psi(\Omega_\omega+\omega)\omega) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}+I)) \\ C(\psi(\Omega_\omega+\psi(\Omega_\omega))\omega) &=& \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2}))) \\ C(\psi(\Omega_\omega+\Omega)) &=& \psi(\Omega_{I2}+\Omega_{I+1}) \\ C(\psi(\Omega_\omega+\Omega_2)) &=& \psi(\Omega_{I2}+\Omega_{I+2}) \\ C(\psi(\Omega_\omega2)) &=& \psi(\Omega_{I2}+\Omega_{I+\omega}) \\ C(\psi(\Omega_\omega2)\omega) &=& \psi(\Omega_{I2}2) \\ C(\psi(\Omega_\omega3)\omega) &=& \psi(\Omega_{I2}3) \\ C(\psi(\Omega_\omega\omega)\omega) &=& \psi(\Omega_{I2}I) \\ C(\psi(\Omega_\omega\Omega)) &=& \psi(\Omega_{I2}\Omega_{I+1}) \\ C(\psi(\Omega_\omega^2)) &=& \psi(\Omega_{I2}\Omega_{I+\omega}) \\ C(\psi(\Omega_\omega^3)) &=& \psi(\Omega_{I2}^2\Omega_{I+\omega}) \\ C(\psi(\varepsilon_{\Omega_\omega+1})) &=& \psi(\varepsilon_{\Omega_{I2}+1}) \\ C(\psi(\Omega_{\omega+1})) &=& \psi(\Omega_{I2+1}) \\ C(\psi(\Omega_{\omega+2})) &=& \psi(\Omega_{I2+2}) \\ C(\psi(\Omega_{\omega2})) &=& \psi(\Omega_{I2+\omega}) \\ C(\psi(\Omega_{\omega2})\omega) &=& \psi(\Omega_{I3}) \\ C(\psi(\Omega_{\omega^2})) &=& \psi(\Omega_{I\omega}) \\ C(\psi(\Omega_{\psi(\Omega)})) &=& \psi(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+1})}) \\ C(\psi(\Omega_{\psi(\Omega_\omega)})) &=& \psi(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+\omega})}) \\ C(\psi(\Omega_{\Omega})) &=& \psi(\Omega_{\Omega_{I+1}}) \\ C(\psi(\Omega_{\Omega_2})) &=& \psi(\Omega_{\Omega_{I+2}}) \\ C(C(1))=C(\psi(\Omega_{\Omega_\omega})) &=& \psi(\Omega_{\Omega_{I+\omega}}) \\ C(\psi(\Omega_{\Omega_\omega})\omega) &=& \psi(\Omega_{\Omega_{I2}}) \\ C(\psi(\Omega_{\Omega_\Omega})) &=& \psi(\Omega_{\Omega_{\Omega_{I+1}}}) \\ C(C(2))=C(\psi(\Omega_{\Omega_{\Omega_\omega}})) &=& \psi(\Omega_{\Omega_{\Omega_{I+\omega}}}) \\ C(C(\omega))=C(\psi(\psi_I(0))) &=& \psi(\psi_{I_2}(0)) \end{eqnarray*}

From \(C(C(\omega))\) to \(C(C(\omega)\omega)\)
First get \(C(C(\omega)+1)\), which is also \(C(\psi(\psi_I(0))+1)\).

\begin{eqnarray*} \text{FGH} & & \text{SGH Ordinal} \\ f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)) \\ f_{\psi(\psi_{I_2}(0)[n])}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_1(\psi_{I_2}(0))) \\ f_{\psi(\psi_{I_2}(0)[n]+\Omega)[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\Omega_2) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_I(0))[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_I(0)) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_I(\Omega_{I+n}))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_I(\Omega_{I+\omega})) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_I(\Omega_{\Omega_{I+n}}))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_I(\Omega_{\Omega_{I+\omega}})) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_I(\psi_{I_2}(0)[n]))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_I(\psi_{I_2}(0))) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_I(\psi_{I_2}(0)[n]+\psi_I(\psi_{I_2}(0)[n])))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_I(\psi_{I_2}(0)+\psi_I(\psi_{I_2}(0)))) \\ f_{\psi(\psi_{I_2}(0)[n]+I)[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+I) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+1}))[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_{\Omega_{I+1}}(\Omega_{I+1})) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_{\Omega_{I+1}}(\Omega_{I+\omega})) \\ f_{\psi(\psi_{I_2}(0)[n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)[n]))}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\psi_{\Omega_{I+1}}(\psi_{I_2}(0))) \\ f_{\psi(\psi_{I_2}(0)[n]+\Omega_{I+1})[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\Omega_{I+1}) \\ f_{\psi(\psi_{I_2}(0)[n]+\Omega_{I+n})}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\Omega_{I+\omega}) \\ f_{\psi(\psi_{I_2}(0)[n]+\Omega_{\Omega_{I+n}})}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)+\Omega_{\Omega_{I+\omega}}) \\ f_{\psi(\psi_{I_2}(0)[n]2)}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)2) \\ f_{\psi(\psi_{I_2}(0)[n]^2)}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(0)^2) \\ f_{\psi(\Omega_{\psi_{I_2}(0)[n]+1})[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\Omega_{\psi_{I_2}(0)+1}) \\ f_{\psi(\Omega_{\psi_{I_2}(0)[n]+I})[n]}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\Omega_{\psi_{I_2}(0)+I}) \\ f_{\psi(\psi_{I_2}(0)[n+1])}f_{\psi(\psi_{I_2}(0))}(n)= & & \\ f_{\psi(\Omega_{\psi_{I_2}(0)[n]2})}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\Omega_{\psi_{I_2}(0)2}) \\ f_{\psi(\psi_{I_2}(0)[n+2])}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\Omega_{\Omega_{\psi_{I_2}(0)2}}) \\ f_{\psi(\psi_{I_2}(0)[2n])}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(1)) \\ f_{\psi(\psi_{I_2}(0)[3n])}f_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(2)) \\ f^2_{\psi(\psi_{I_2}(0))}(n) & & \psi(\psi_{I_2}(\psi(\psi_{I_2}(0)))) \\ f_{\psi(\psi_{I_2}(0))+1}(n) & & \psi(\psi_{I_2}(\Omega)) \\ f_{\psi(\psi_{I_2}(\Omega))}(n) & & \psi(\psi_{I_2}(\Omega_2)) \\ f_{\psi(\psi_{I_2}(\Omega_\omega))}(n) & & \psi(\psi_{I_2}(\Omega_\omega)) \end{eqnarray*}

Then continue to \(C(C(\omega)\omega)=C(\psi(\psi_I(0))\omega)\). This is very simple. \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\psi_I(0))+1) &=& \psi(\psi_{I_2}(\Omega_\omega)) \\ C(\psi(\psi_I(0))+\omega) &=& \psi(\psi_{I_2}(\psi_I(0))) \\ C(\psi(\psi_I(0))2) &=& \psi(\psi_{I_2}(\psi_I(\psi_{I_2}(0)))) \\ C(\psi(\psi_I(0))3) &=& \psi(\psi_{I_2}(\psi_I(\psi_{I_2}(\psi_I(\psi_{I_2}(0)))))) \\ C(\psi(\psi_I(0))\omega) &=& \psi(\psi_{I_2}(I)) \end{eqnarray*}

From \(C(C(\omega)\omega)\) to \(C(\psi(\psi_I(1)))\)
First we should get \(C(\psi(\psi_I(0))\omega+1)\). Here you may find some patterns. Notice again that \(\psi(\psi_{I_2}(I)[n+1])=\psi(\psi_{I_2}(\psi_I(\psi_{I_2}(I)[n])))=\psi(\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)[n]))))\).

\begin{eqnarray*} \text{FGH} & & \text{SGH Ordinal} \\ f_{\psi(\psi_{I_2}(I)[n])}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_1(\psi_{I_2}(I))) \\ f_{\psi(\psi_{I_2}(I)[n]+\Omega)[n]}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\Omega_2) \\ f_{\psi(\psi_{I_2}(I)[n]+I)[n]}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+I) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+1}))[n]}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+1})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\omega})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(0)[n]}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(0)})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+n})}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I+\omega})})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})[n]}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\Omega_{I2})})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(0))[n]}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(0))})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(I)[n])}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(I))})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(I)[n]+I)}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(I)+I)})) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+\psi_I(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}( \\ & & \Omega_{I+\psi_I(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I+\omega}))})) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\Omega_{I2}))[n]}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\Omega_{I2})) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)[n]))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(0))) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{I2})[n])))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{I2})))) \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(0)[n]))))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(0))))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[n+1])}f_{\psi(\psi_{I_2}(I))}(n)= & & \\ f_{\psi(\psi_{I_2}(I)[n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)[n]))))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I))))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[n+2])}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I) \\ & & +\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)))))))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[2n])}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[2n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)[n]))}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)) \\ & & +\psi_{\Omega_{I+1}}(\psi_{I_2}(0))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[2n+1])}f_{\psi(\psi_{I_2}(I))}(n)= & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)) \\ f_{\psi(\psi_{I_2}(I)[2n]+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)[2n]))))}f_{\psi(\psi_{I_2}(I))}(n) & & +\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)))))) \\ --- & & --- \\ f_{\psi(\psi_{I_2}(I)[3n])}f_{\psi(\psi_{I_2}(I))}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))2) \\ f_{\psi(\psi_{I_2}(I))+1}(n) & & \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\Omega) \\ C(\psi(\psi_I(0))\omega+1) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\Omega_\omega) \end{eqnarray*}

And then...

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\psi_I(0))\omega+\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\psi_I(0)) \\ C(\psi(\psi_I(0))\omega^2) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))I) \\ C(\psi(\psi_I(0))\omega^\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))I^\omega) \\ C(\psi(\psi_I(0))\varepsilon_0) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\varepsilon_{I+1}) \\ C(\psi(\psi_I(0))\psi(\Omega)) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\psi_{\Omega_{I+1}}(\Omega_{I+1})) \\ C(\psi(\psi_I(0))\psi(\Omega_\omega)\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\psi_{\Omega_{I+1}}(\Omega_{I2})) \\ C(\psi(\psi_I(0))\psi(\Omega_{\Omega_\omega})) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\psi_{\Omega_{I+1}}(\Omega_{\Omega_{I+\omega}})) \\ C(\psi(\psi_I(0))^2) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\psi_{\Omega_{I+1}}(\psi_{I_2}(0))) \\ C(\psi(\psi_I(0))^2\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))^2) \\ C(\psi(\psi_I(0))^\omega\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))^I) \\ C(\psi(\psi_I(0))^{\psi(\psi_I(0))}\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))^{\psi_{\Omega_{I+1}}(\psi_{I_2}(I))}) \\ C(\psi(\psi_I(0)+1)) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)+1)) \\ C(\psi(\psi_I(0)+\psi(\psi_I(0)))\omega) &=& \psi(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)))) \\ C(\psi(\psi_I(0)+\Omega)) &=& \psi(\psi_{I_2}(I)+\Omega_{I+1}) \\ C(\psi(\psi_I(0)+\Omega_2)) &=& \psi(\psi_{I_2}(I)+\Omega_{I+2}) \\ C(\psi(\psi_I(0)+\Omega_\omega)\omega) &=& \psi(\psi_{I_2}(I)+\Omega_{I2}) \\ C(\psi(\psi_I(0)2)) &=& \psi(\psi_{I_2}(I)+\psi_{I_2}(0)) \\ C(\psi(\psi_I(0)2)\omega) &=& \psi(\psi_{I_2}(I)2) \\ C(\psi(\psi_I(0)\omega)) &=& \psi(\psi_{I_2}(I)\omega) \\ C(\psi(\psi_I(0)^2)) &=& \psi(\psi_{I_2}(I)\psi_{I_2}(0)) \\ C(\psi(\psi_I(0)^2)\omega) &=& \psi(\psi_{I_2}(I)^2) \\ C(\psi(\psi_I(0)^{\psi_I(0)})\omega) &=& \psi(\psi_{I_2}(I)^{\psi_{I_2}(I)}) \\ C(\psi(\varepsilon_{\psi_I(0)+1})) &=& \psi(\varepsilon_{\psi_{I_2}(I)+1}) \\ C(\psi(\Omega_{\psi_I(0)+1})) &=& \psi(\Omega_{\psi_{I_2}(I)+1}) \\ C(\psi(\Omega_{\psi_I(0)+2})) &=& \psi(\Omega_{\psi_{I_2}(I)+2}) \\ C(\psi(\Omega_{\psi_I(0)+\omega})\omega) &=& \psi(\Omega_{\psi_{I_2}(I)+I}) \\ C(\psi(\Omega_{\psi_I(0)+\omega+1})) &=& \psi(\Omega_{\psi_{I_2}(I)+I+1}) \\ C(\psi(\Omega_{\psi_I(0)2})) &=& \psi(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(0)}) \\ C(\psi(\Omega_{\psi_I(0)2})\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}) \end{eqnarray*}

Then we should get \(C(\psi(\Omega_{\psi_I(0)2})\omega+1)\) first. Having gone so far, you can see how the catching ordinals work. This time, please notice that \(\psi(\Omega_{\psi_{I_2}(I)2[n+1]})=\psi(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)2[n]}))})\). Basicly \(\psi(\Omega_{\psi_{I_2}(I)2[n]})=\psi(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(I)[n]})\), not \(\psi(\Omega_{\psi_{I_2}(I)[n]2})\).

\begin{eqnarray*} \text{FGH} & & \text{SGH Ordinal} \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]})}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_1(\Omega_{\psi_{I_2}(I)2})) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\Omega)[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_2) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_I(0))[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_I(0)) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_I(\Omega_{\psi_{I_2}(I)2[n]}))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_I(\Omega_{\psi_{I_2}(I)2})) \\ --- & & --- \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_I(\Omega_{\psi_{I_2}(I)2[n]}+\psi_I(\Omega_{\psi_{I_2}(I)2[n]})))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_I(\Omega_{\psi_{I_2}(I)2} \\ & & +\psi_I(\Omega_{\psi_{I_2}(I)2}))) \\ --- & & --- \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+I)[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+I) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\varepsilon_{I+1}[n])}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\varepsilon_{I+1}) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\Omega_{I+n}))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{I+\omega})) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)[n]))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(0))) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I)[n]))))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\psi_{I_2}(I))))) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)+1})[n])))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)+1})))) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)2[n]}))))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)2})))) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)))[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\Omega_n)}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I))\Omega_\omega) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)+1))[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\psi_{I_2}(I)+1)) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)+1}))[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)+1})) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(\psi_I(0))}))[n]}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(\psi_I(0))})) \\ --- & & --- \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n+1]})}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n)= & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}( \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[n]}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)+\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)2[n]}))}))}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \Omega_{\psi_{I_2}(I)+\psi_{I_2}(\psi_I(\Omega_{\psi_{I_2}(I)2}))})) \\ --- & & --- \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[2n]})}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2[3n]})}f_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})2) \\ --- & & --- \\ f^2_{\psi(\Omega_{\psi_{I_2}(I)2})}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2}) \\ & & \times\psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2}))) \\ --- & & --- \\ f_{\psi(\Omega_{\psi_{I_2}(I)2})+1}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})\Omega) \\ f_{\psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})\Omega_\omega)}(n) & & \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})\Omega_\omega) \end{eqnarray*}

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\Omega_{\psi_I(0)2})\omega+1) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})\Omega_\omega) \\ C(\psi(\Omega_{\psi_I(0)2})\omega^2) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})I) \\ C(\psi(\Omega_{\psi_I(0)2})\psi(\Omega)) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})\psi_{\Omega_{I+1}}(\Omega_{I+1})) \\ C(\psi(\Omega_{\psi_I(0)2})^2\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})^2) \\ C(\psi(\Omega_{\psi_I(0)2})^\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2})^\omega) \\ C(\psi(\Omega_{\psi_I(0)2}+1)) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{I+1}}(\Omega_{\psi_{I_2}(I)2}+1)) \\ C(\psi(\Omega_{\psi_I(0)2}+\Omega)) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_{I+1}) \\ C(\psi(\Omega_{\psi_I(0)2}+\Omega_\omega)) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_{I+\omega}) \\ C(\psi(\Omega_{\psi_I(0)2}+\psi_I(0))) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{I_2}(0)) \\ C(\psi(\Omega_{\psi_I(0)2}+\varepsilon_{\psi_I(0)+1})) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\varepsilon_{\psi_{I_2}(I)+1}) \\ C(\psi(\Omega_{\psi_I(0)2}+\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}))) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{\psi_{I_2}(I)+1}}(\Omega_{\psi_{I_2}(I)+1})) \\ C(\psi(\Omega_{\psi_I(0)2}+\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)2}))\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\psi_{\Omega_{\psi_{I_2}(I)+1}}(\Omega_{\psi_{I_2}(I)2})) \\ C(\psi(\Omega_{\psi_I(0)2}+\Omega_{\psi_I(0)+1})) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_{\psi_{I_2}(I)+1}) \\ C(\psi(\Omega_{\psi_I(0)2}+\Omega_{\psi_I(0)+\omega})) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_{\psi_{I_2}(I)+\omega}) \\ C(\psi(\Omega_{\psi_I(0)2}2)) &=& \psi(\Omega_{\psi_{I_2}(I)2}+\Omega_{\psi_{I_2}(I)+\psi_{I_2}(0)}) \\ C(\psi(\Omega_{\psi_I(0)2}2)\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}2) \\ C(\psi(\Omega_{\psi_I(0)2}\omega)) &=& \psi(\Omega_{\psi_{I_2}(I)2}\omega) \\ C(\psi(\Omega_{\psi_I(0)2}^2)\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2}^2) \\ C(\psi(\Omega_{\psi_I(0)2+1})) &=& \psi(\Omega_{\psi_{I_2}(I)2+1}) \\ C(\psi(\Omega_{\psi_I(0)2+1})\omega) &=& \psi(\Omega_{\psi_{I_2}(I)2+I}) \\ C(\psi(\Omega_{\psi_I(0)3})\omega) &=& \psi(\Omega_{\psi_{I_2}(I)3}) \\ C(\psi(\Omega_{\psi_I(0)\omega})) &=& \psi(\Omega_{\psi_{I_2}(I)\omega}) \\ C(\psi(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(0)})) &=& \psi(\Omega_{\psi_{\Omega_{\psi_{I_2}(I)+1}}(0)}) \\ C(\psi(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)2})})\omega) &=& \psi(\Omega_{\psi_{\Omega_{\psi_{I_2}(I)+1}}(\Omega_{\psi_{I_2}(I)2})}) \\ C(\psi(\Omega_{\Omega_{\psi_I(0)+1}})) &=& \psi(\Omega_{\Omega_{\psi_{I_2}(I)+1}}) \\ C(\psi(\Omega_{\Omega_{\psi_I(0)2}})) &=& \psi(\Omega_{\Omega_{\psi_{I_2}(I)+\psi_{I_2}(0)}}) \\ C(\psi(\Omega_{\Omega_{\Omega_{\psi_I(0)+1}}})) &=& \psi(\Omega_{\Omega_{\Omega_{\psi_{I_2}(I)+1}}}) \\ C(\psi(\psi_I(1))) &=& \psi(\psi_{I_2}(I+1)) \end{eqnarray*}

From \(C(\psi(\psi_I(1)))\) to \(C(C(\omega^2))\)
The \(C(C(\omega^2))=C(\psi(I))\) since \(C(\omega^2)=\psi(I)\). \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\psi_I(1))+1) &=& \psi(\psi_{I_2}(I+\Omega_\omega)) \\ C(\psi(\psi_I(1))2) &=& \psi(\psi_{I_2}(I+\psi_I(\psi_{I_2}(I+1)))) \\ C(\psi(\psi_I(1))\omega) &=& \psi(\psi_{I_2}(I2)) \\ C(\psi(\psi_I(1))\omega+1) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))\Omega_\omega) \\ C(\psi(\psi_I(1))\omega^2) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))I) \\ C(\psi(\psi_I(1))\omega^\omega) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))I^\omega) \\ C(\psi(\psi_I(1))^2) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))\psi_{\Omega_{I+1}}(\psi_{I_2}(I+1))) \\ C(\psi(\psi_I(1))^2\omega) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))^2) \\ C(\psi(\psi_I(1))^\omega) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))^\omega) \\ C(\psi(\psi_I(1))^{\psi(\psi_I(1))}\omega) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))^{\psi_{\Omega_{I+1}}(\psi_{I_2}(I2))}) \\ C(\psi(\psi_I(1)+1)) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2)+1)) \\ C(\psi(\psi_I(1)+\psi(\psi_I(1)))) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I+1)))) \\ C(\psi(\psi_I(1)+\psi(\psi_I(1)+1))) &=& \psi(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2)+\psi_{\Omega_{I+1}}(\psi_{I_2}(I2)+1))) \\ C(\psi(\psi_I(1)+\Omega)) &=& \psi(\psi_{I_2}(I2)+\Omega_{I+1}) \\ C(\psi(\psi_I(1)2)\omega) &=& \psi(\psi_{I_2}(I2)2) \\ C(\psi(\psi_I(1)^2)\omega) &=& \psi(\psi_{I_2}(I2)^2) \\ C(\psi(\varepsilon_{\psi_I(1)+1})) &=& \psi(\varepsilon_{\psi_{I_2}(I2)+1}) \\ C(\psi(\Omega_{\psi_I(1)+1})) &=& \psi(\Omega_{\psi_{I_2}(I2)+1}) \\ C(\psi(\psi_I(2))) &=& \psi(\psi_{I_2}(I2+1)) \\ C(\psi(\psi_I(2))\omega) &=& \psi(\psi_{I_2}(I3)) \\ C(\psi(\psi_I(\omega))) &=& \psi(\psi_{I_2}(I\omega)) \\ C(\psi(\psi_I(\psi(\Omega)))) &=& \psi(\psi_{I_2}(\psi_{\Omega_{I+1}}(\Omega_{I+1}))) \\ C(\psi(\psi_I(\Omega))) &=& \psi(\psi_{I_2}(\Omega_{I+1})) \\ C(C(\omega+1))=C(\psi(\psi_I(\Omega_\omega))) &=& \psi(\psi_{I_2}(\Omega_{I+\omega})) \\ C(C(\omega+2))=C(\psi(\psi_I(\Omega_{\Omega_\omega}))) &=& \psi(\psi_{I_2}(\Omega_{\Omega_{I+\omega}})) \\ C(C(\omega2))=C(\psi(\psi_I(\psi_I(0)))) &=& \psi(\psi_{I_2}(\psi_{I_2}(0))) \\ C(C(\omega3))=C(\psi(\psi_I(\psi_I(\psi_I(0))))) &=& \psi(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0)))) \\ C(C(\omega^2))=C(\psi(I)) &=& \psi(I_2) \end{eqnarray*}

From \(C(C(\omega^2))\) to \(C(C(C(\omega)))\)
\(\omega\)'s inside Catching function work as I's in normal notation; \(\Omega\)'s in C(ψ) work as \(\Omega_{I+1}\)'s in normal notation; and \(\Omega_\omega\)'s in C(ψ) work as \(\Omega_{I2}\)'s in normal notation. Here you'll see, I's in C(ψ) work as \(I_2\)'s in normal notation.

From \(C(C(\omega^2))\) to \(C(C(\varepsilon_0))\)
Now I must tell you how to compare FGH to SGH from "inside" - e.g. compare them from \(C(\psi(I))\) to \(C(\psi(I)+1)\):

In SGH the "base number" never changes, so the length of all the fundamental sequences we need is always n. SGH is sensitive with fundamental sequence but not sensitive with +1's (i.e. successors) compared with FGH. e.g. \(g_{\omega^{\varepsilon_0}}(10)=g_{\omega^{\varepsilon_0[10]}}(10)=g_{\omega^{\omega\text{^^}10}}(10)=10\uparrow\uparrow11=g_{\varepsilon_0[11]}(10)\) and \(g_{\varepsilon_0+1}(10)=g_{\varepsilon_0}(10)+1=g_{\omega\text{^^}10}(10)+1=10\uparrow\uparrow10+1\), but actually \(\omega^{\varepsilon_0}=\varepsilon_0<\varepsilon_0+1\).

\(f_{\psi(I_2)}(n)=f_{\psi(I_2)[n]}(n)=f_{\psi(\psi_{I_2}(I_2)[n])}(n)\approx g_{\psi(\psi_{I_2}(I_2)[n])}(n)=g_{\psi(I_2)}(n)\). The next thing in FGH is \(f_{\psi(I_2)}(n+1)\), and we need \(f_{\psi(I_2)[n+1]}(n+1)=f_{\psi(\psi_{I_2}(I_2)[n+1])}(n+1)=f_{\psi(\psi_{I_2}(\psi_{I_2}(I_2)[n]))}(n+1)\) here. We can safely add a prefix "I2+" in SGH, so it becomes \(\psi(\psi_{I_2}(I_2)[n+1])=\psi(\psi_{I_2}(\psi_{I_2}(I_2)[n]))\) \(=\psi(\psi_{I_2}(\psi_{I_2}(I_2)[n-1]+\psi_{I_2}(I_2)[n]))=\psi(\psi_{I_2}(I_2+\psi_{I_2}(I_2)[n]))\) in SGH. Next \(f_{\psi(I_2)}(n+2)=f_{\psi(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(I_2)[n])))}(n+2)\) becomes \(\psi(\psi_{I_2}(I_2+\psi_{I_2}(I_2+\psi_{I_2}(I_2)[n])))\) in SGH. What these lead to is \(f_{\psi(I_2)}(2n)\approx g_{\psi(I_22)}(n)\). If we use ordinal notation in Bird's type (he use notations like \(\theta(\theta_1(\Omega_2^2)+\Omega)\), and \(\theta(\Omega_4)=\theta(\theta_1(\theta_2(\theta_3(\Omega_4))))\), etc.), actually we get \(\psi(\psi_{I_2}(I_22))=\psi(\psi_{I_2}(I_2+\psi_{I_2}(I_2+...\psi_{I_2}(I_2+\psi_{I_2}(I_2))...)))\) in SGH.

Further, \(f_{\psi(I_2)}(3n)\approx g_{\psi(I_23)}(n)\), \(f_{\psi(I_2)}(n^2)\approx g_{\psi(I_2\omega)}(n)\), \(f_{\psi(I_2)}f_3(n)\approx g_{\psi(I_2\varepsilon_0)}(n)\), \(f_{\psi(I_2)}^2(n)\approx g_{\psi(I_2\psi(I_2))}(n)\), \(f_{\psi(I_2)}^3(n)\approx g_{\psi(I_2\psi(I_2\psi(I_2)))}(n)\), \(f_{\psi(I_2)+1}(n)\approx g_{\psi(I_2\Omega)}(n)\),and finally the next catching ordinal is \(\psi(I_2\Omega_\omega)\).

\begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(I)+1) &=& \psi(I_2\Omega_\omega) \\ C(\psi(I)2) &=& \psi(I_2\psi_I(I_2)) \\ C(\psi(I)\omega) &=& \psi(I_2I) \\ C(\psi(I)\omega+1) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)\Omega_\omega) \\ C(\psi(I)\omega^2) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)I) \\ C(\psi(I)\omega^\omega) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)I^\omega) \\ C(\psi(I)^2) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)\psi_{\Omega_{I+1}}(I_2)) \\ C(\psi(I)^2\omega) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)^2) \\ C(\psi(I)^\omega) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I)^\omega) \\ C(\psi(I+1)) &=& \psi(I_2I+\psi_{\Omega_{I+1}}(I_2I+1)) \\ C(\psi(I+\Omega)) &=& \psi(I_2I+\Omega_{I+1}) \\ C(\psi(I+\Omega_\omega)) &=& \psi(I_2I+\Omega_{I+\omega}) \\ C(\psi(I+\psi_I(0))) &=& \psi(I_2I+\psi_{I_2}(0)) \\ C(\psi(I+\psi_I(I))) &=& \psi(I_2I+\psi_{I_2}(I_2)) \\ C(\psi(I2)) &=& \psi(I_2I+I_2) \\ C(\psi(I2)\omega) &=& \psi(I_2I2) \\ C(\psi(I3)) &=& \psi(I_2I2+I_2) \\ C(\psi(I\omega)) &=& \psi(I_2I\omega) \\ C(\psi(I\omega)\omega) &=& \psi(I_2I^2) \\ C(\psi(I\omega^\omega)) &=& \psi(I_2I^\omega) \\ C(\psi(I\psi(I))) &=& \psi(I_2\psi_{\Omega_{I+1}}(I_2)) \\ C(\psi(I\Omega)) &=& \psi(I_2\Omega_{I+1}) \\ C(C(\omega^2+\omega))=C(\psi(I\psi_I(0))) &=& \psi(I_2\psi_{I_2}(0)) \\ C(C(\omega^22))=C(\psi(I\psi_I(I))) &=& \psi(I_2\psi_{I_2}(I_2)) \\ C(C(\omega^3))=C(\psi(I^2)) &=& \psi(I_2^2) \\ C(C(\omega^\omega))=C(\psi(I^\omega)) &=& \psi(I_2^\omega) \\ C(\psi(I^\omega)\omega) &=& \psi(I_2^I) \\ C(C(\omega^{\omega+1}))=C(\psi(I^I)) &=& \psi(I_2^{I_2}) \\ C(C(\omega^{\omega+2}))=C(\psi(I^{I+1})) &=& \psi(I_2^{I_2+1}) \\ C(C(\omega^{\omega2+1}))=C(\psi(I^{I2})) &=& \psi(I_2^{I_22}) \\ C(C(\omega^{\omega^2+1}))=C(\psi(I^{I^2})) &=& \psi(I_2^{I_2^2}) \\ C(C(\omega^{\omega^\omega+1}))=C(\psi(I^{I^I})) &=& \psi(I_2^{I_2^{I_2}}) \\ C(C(\omega^{\omega^{\omega^\omega}+1}))=C(\psi(I^{I^{I^I}})) &=& \psi(I_2^{I_2^{I_2^{I_2}}}) \\ C(C(\varepsilon_0))=C(\psi(\varepsilon_{I+1})) &=& \psi(\varepsilon_{I_2+1}) \end{eqnarray*}

From \(C(C(\varepsilon_0))\) to \(C(C(C(\omega)))\)
Something interesting: the \(\Omega_{I+1}\) in \(C(\psi)\) works as \(\Omega_{I_2+1}\) in \(\psi\) function, the \(\Omega_{I+2}\) as \(\Omega_{I_2+2}\), and the \(\Omega_{I+\omega}\) as \(\Omega_{I_2+I}\), etc. \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\varepsilon_{I+1})\omega) &=& \psi(\varepsilon_{I_2+I}) \\ C(\psi(\varepsilon_{I+1})\omega^2) &=& \psi(\varepsilon_{I_2+I}+\psi_{\Omega_{I+1}}(\varepsilon_{I_2+I})I) \\ C(\psi(\varepsilon_{I+1})^2\omega) &=& \psi(\varepsilon_{I_2+I}+\psi_{\Omega_{I+1}}(\varepsilon_{I_2+I})^2) \\ C(\psi(\varepsilon_{I+1}+1)) &=& \psi(\varepsilon_{I_2+I}+\psi_{\Omega_{I+1}}(\varepsilon_{I_2+I}+1)) \\ C(\psi(\varepsilon_{I+1}+\Omega)) &=& \psi(\varepsilon_{I_2+I}+\Omega_{I+1}) \\ C(\psi(\varepsilon_{I+1}+\Omega_\omega)\omega) &=& \psi(\varepsilon_{I_2+I}+\Omega_{I2}) \\ C(\psi(\varepsilon_{I+1}+\psi_I(0))) &=& \psi(\varepsilon_{I_2+I}+\psi_{I_2}(0)) \\ C(\psi(\varepsilon_{I+1}+\psi_I(I))) &=& \psi(\varepsilon_{I_2+I}+\psi_{I_2}(I_2)) \\ C(\psi(\varepsilon_{I+1}+\psi_I(\varepsilon_{I+1}))) &=& \psi(\varepsilon_{I_2+I}+\psi_{I_2}(\varepsilon_{I_2+1})) \\ C(\psi(\varepsilon_{I+1}+I)) &=& \psi(\varepsilon_{I_2+I}+I_2) \\ C(\psi(\varepsilon_{I+1}+I^I)) &=& \psi(\varepsilon_{I_2+I}+I_2^{I_2}) \\ C(\psi(\varepsilon_{I+1}2)) &=& \psi(\varepsilon_{I_2+I}+\varepsilon_{I_2+1}) \\ C(\psi(\varepsilon_{I+1}\omega)) &=& \psi(\varepsilon_{I_2+I}\omega) \\ C(\psi(\varepsilon_{I+1}I)) &=& \psi(\varepsilon_{I_2+I}I_2) \\ C(\psi(\varepsilon_{I+1}^2)\omega) &=& \psi(\varepsilon_{I_2+I}^2) \\ C(\psi(\varepsilon_{I+2})) &=& \psi(\varepsilon_{I_2+I+1}) \\ C(\psi(\varepsilon_{I+2})\omega) &=& \psi(\varepsilon_{I_2+I2}) \\ C(\psi(\varepsilon_{I+\omega})\omega) &=& \psi(\varepsilon_{I_2+I^2}) \\ C(\psi(\varepsilon_{I+\Omega})) &=& \psi(\varepsilon_{I_2+\Omega_{I+1}}) \\ C(\psi(\varepsilon_{I+\Omega_\omega})) &=& \psi(\varepsilon_{I_2+\Omega_{I+\omega}}) \\ C(\psi(\varepsilon_{I+\psi_I(0)})) &=& \psi(\varepsilon_{I_2+\psi_{I_2}(0)}) \\ C(\psi(\varepsilon_{I+\psi_I(\varepsilon_{I+1})})) &=& \psi(\varepsilon_{I_2+\psi_{I_2}(\varepsilon_{I_2+1})}) \\ C(\psi(\varepsilon_{I2})) &=& \psi(\varepsilon_{I_22}) \\ C(\psi(\varepsilon_{I3})) &=& \psi(\varepsilon_{I_23}) \\ C(\psi(\varepsilon_{I^2})) &=& \psi(\varepsilon_{I_2^2}) \\ C(\psi(\varepsilon_{\varepsilon_{I+1}})) &=& \psi(\varepsilon_{\varepsilon_{I_2+1}}) \\ C(\psi(\Omega_{I+1})) &=& \psi(\Omega_{I_2+1}) \\ C(\psi(\Omega_{I+\omega})\omega) &=& \psi(\Omega_{I_2+I}) \\ C(\psi(\Omega_{I+\Omega_\omega})) &=& \psi(\Omega_{I_2+\Omega_{I+\omega}}) \\ C(\psi(\Omega_{I+\psi_I(\Omega_{I+1})})) &=& \psi(\Omega_{I_2+\psi_{I_2}(\Omega_{I_2+1})}) \\ C(\psi(\Omega_{I2})) &=& \psi(\Omega_{I_22}) \\ C(\psi(\Omega_{I3})) &=& \psi(\Omega_{I_23}) \\ C(\psi(\Omega_{\Omega_{I+1}})) &=& \psi(\Omega_{\Omega_{I_2+1}}) \\ C(\psi(\psi_{I_2}(0))) &=& \psi(\psi_{I_3}(0)) \end{eqnarray*} So \(C(C(C(\omega)))=C(C(\psi(\psi_I(0))))=C(\psi(\psi_{I_2}(0)))=\psi(\psi_{I_3}(0))\).

From \(C(C(C(\omega)))\) to \(C(\Omega)\)
The \(I_2\) in \(C(\psi)\) works as \(I_3\) in \(\psi\) function, the \(I_3\) as \(I_4\), etc. so it's not hard to image the strength of the Catching function. \begin{eqnarray*} \text{Catching function} & & \text{Normal notation} \\ C(\psi(\psi_{I_2}(0))+1) &=& \psi(\psi_{I_3}(\Omega_\omega)) \\ C(\psi(\psi_{I_2}(0))\omega) &=& \psi(\psi_{I_3}(I)) \\ C(\psi(\psi_{I_2}(0))\omega^2) &=& \psi(\psi_{I_3}(I)+\psi_{\Omega_{I+1}}(\psi_{I_3}(I))I) \\ C(\psi(\psi_{I_2}(0))^2\omega) &=& \psi(\psi_{I_3}(I)+\psi_{\Omega_{I+1}}(\psi_{I_3}(I))^2) \\ C(\psi(\psi_{I_2}(0)+1)) &=& \psi(\psi_{I_3}(I)+\psi_{\Omega_{I+1}}(\psi_{I_3}(I)+1)) \\ C(\psi(\psi_{I_2}(0)+\Omega)) &=& \psi(\psi_{I_3}(I)+\Omega_{I+1}) \\ C(\psi(\psi_{I_2}(0)+\Omega_\Omega)) &=& \psi(\psi_{I_3}(I)+\Omega_{\Omega_{I+1}}) \\ C(\psi(\psi_{I_2}(0)+\psi_I(0))) &=& \psi(\psi_{I_3}(I)+\psi_{I_2}(0)) \\ C(\psi(\psi_{I_2}(0)+\psi_I(\psi_{I_2}(0)))\omega) &=& \psi(\psi_{I_3}(I)+\psi_{I_2}(\psi_{I_3}(I))) \\ C(\psi(\psi_{I_2}(0)+I)) &=& \psi(\psi_{I_3}(I)+I_2) \\ C(\psi(\psi_{I_2}(0)+\varepsilon_{I+1})) &=& \psi(\psi_{I_3}(I)+\varepsilon_{I_2+1}) \\ C(\psi(\psi_{I_2}(0)+\psi_{\Omega_{I+1}}(\Omega_{I+1}))) &=& \psi(\psi_{I_3}(I)+\psi_{\Omega_{I_2+1}}(\Omega_{I_2+1})) \\ C(\psi(\psi_{I_2}(0)+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)))) &=& \psi(\psi_{I_3}(I)+\psi_{\Omega_{I_2+1}}(\psi_{I_3}(0))) \\ C(\psi(\psi_{I_2}(0)+\Omega_{I+1})) &=& \psi(\psi_{I_3}(I)+\Omega_{I_2+1}) \\ C(\psi(\psi_{I_2}(0)+\Omega_{I+2})) &=& \psi(\psi_{I_3}(I)+\Omega_{I_2+I+1}) \\ C(\psi(\psi_{I_2}(0)+\Omega_{I2})) &=& \psi(\psi_{I_3}(I)+\Omega_{I_22}) \\ C(\psi(\psi_{I_2}(0)+\Omega_{\Omega_{I+1}})) &=& \psi(\psi_{I_3}(I)+\Omega_{\Omega_{I_2+1}}) \\ C(\psi(\psi_{I_2}(0)2)) &=& \psi(\psi_{I_3}(I)+\psi_{I_3}(0)) \\ C(\psi(\psi_{I_2}(0)2)\omega) &=& \psi(\psi_{I_3}(I)2) \\ C(\psi(\psi_{I_2}(0)\omega)\omega) &=& \psi(\psi_{I_3}(I)I) \\ C(\psi(\psi_{I_2}(0)I)) &=& \psi(\psi_{I_3}(I)I_2) \\ C(\psi(\psi_{I_2}(0)\Omega_{I+1})) &=& \psi(\psi_{I_3}(I)\Omega_{I_2+1}) \\ C(\psi(\psi_{I_2}(0)^2)\omega) &=& \psi(\psi_{I_3}(I)^2) \\ C(\psi(\psi_{I_2}(0)^{\psi_{I_2}(0)})\omega) &=& \psi(\psi_{I_3}(I)^{\psi_{I_3}(I)}) \\ C(\psi(\varepsilon_{\psi_{I_2}(0)+1})) &=& \psi(\varepsilon_{\psi_{I_3}(I)+1}) \\ C(\psi(\Omega_{\psi_{I_2}(0)+1})) &=& \psi(\Omega_{\psi_{I_3}(I)+1}) \\ C(\psi(\Omega_{\psi_{I_2}(0)2})\omega) &=& \psi(\Omega_{\psi_{I_3}(I)2}) \\ C(\psi(\psi_{I_2}(1))) &=& \psi(\psi_{I_3}(I+1)) \\ C(\psi(\psi_{I_2}(I))) &=& \psi(\psi_{I_3}(I_2)) \\ C(\psi(\psi_{I_2}(\psi_{I_2}(0)))) &=& \psi(\psi_{I_3}(\psi_{I_3}(0))) \\ C(\psi(I_2)) &=& \psi(I_3) \\ C(\psi(I_2I)) &=& \psi(I_3I_2) \\ C(\psi(I_2^2)) &=& \psi(I_3^2) \\ C(\psi(\Omega_{I_2+1})) &=& \psi(\Omega_{I_3+1}) \\ C(\psi(\Omega_{I_2+I})) &=& \psi(\Omega_{I_3+I_2}) \\ C(\psi(\Omega_{I_22})) &=& \psi(\Omega_{I_32}) \\ C(\psi(\Omega_{\Omega_{I_2+1}})) &=& \psi(\Omega_{\Omega_{I_3+1}}) \\ C(\psi(\psi_{I_3}(0))) &=& \psi(\psi_{I_4}(0)) \\ C(\psi(\psi_{I_3}(I))) &=& \psi(\psi_{I_4}(I_2)) \\ C(\psi(\psi_{I_3}(I_2))) &=& \psi(\psi_{I_4}(I_3)) \\ C(\psi(I_3)) &=& \psi(I_4) \\ C(\psi(I_4)) &=& \psi(I_5) \\ C(\Omega) &=& \psi(\psi_{I_\omega}(0)) \end{eqnarray*} This has comparable growth rate to the ? function in HAN, also {L,X,2}n,n in BEAF. So we know the BIGG falls below an L2 structure. Next we'll see that BIGG falls FAR below an L2 structure.

=Catching Hierarchy I= Catching hierarchy I range from \(C(\Omega)\) to \(C(\alpha)\), where \(\alpha\) is a large 1-uncountable ordinal.

From \(C(\Omega)\) to \(C(\Omega^2)\)
From now on I use R function for ordinal notation. {0} is \(\omega\) - the smallest limit ordinal. All the rules except these two still work, because they don't change the base number n: nR0=10^n, nRa+1=nRaRa...Ra with n Ra's. Okey, I would call these rules "work on strings only". Here the string is everything after the R symbol. So {0,1} is \(\varepsilon_0\), {0,2} is BHO, {0,1{0}} is TFB and {0,0,{0}} is \(\psi(\psi_{I_\omega}(0))\). This notation goes as far as R function.

And notice that R-function-look ordinal notation doesn't use uncountable ordinals for collasping, so all the notations mean countable ordinals.

From \(C(\Omega)\) to \(C(\Omega+\psi(\psi_{I_\omega}(1)))\)
{0,0,{0,0,1}}{0,0,{0,0,1}} means \(\psi(\psi_{I_I}(0)2)=\psi(\psi_{I_I}(0)+\psi_{I_I}(0))=\psi(\alpha\mapsto\psi_I(\psi_{I_I}(0)+\psi_{I_\alpha}(0)))\). If this ordinal is in SGH, it'll be equvalent to \(f_{\psi(\psi_{I_I}(0))}(2n)\), or (2n)R{0,0,{0,0,1}}.

And then \(\psi(\psi_{I_I}(0)\Omega_\omega)\) will be the next catching ordinal.

From \(C(\zeta_{\Omega+1})\) to \(C(\vartheta_1(\Omega_\omega))\)
=Catching Hierarchy II= Catching hierarchy II range from \(C(C_1(0))\) to \(C(\chi(\alpha))\), where \(\alpha\) is a large mahlo ordinal. It's enough for BEAF.