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}) \end{eqnarray*}

\begin{eqnarray*}

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(\varepsilon_{\Omega_{I+1}+I}\Omega_\omega)=\psi(\psi_{\Omega_{I+2}}(I)\Omega_\omega) \\ C(\psi(\varepsilon_{\Omega+1})\omega^2) &=& \psi(\varepsilon_{\Omega_{I+1}+I}I)=\psi(\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}}(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)\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I)+\Omega_{I+1})) \\ C(\psi(\varepsilon_{\Omega+1}2)) &=& \psi(\psi_{\Omega_{I+2}}(I)\psi_{\Omega_{I+1}}(\psi_{\Omega_{I+2}}(I)+\psi_{\Omega_{I+2}}(0))) \\ C(\psi(\varepsilon_{\Omega+1}\omega)) &=& \psi(\psi_{\Omega_{I+2}}(I)\psi_{\Omega_{I+1}}(\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_{I2}[n]))}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\psi_{\Omega_{I+1}}(\Omega_{I2})) \\ f_{\psi(\Omega_{I2}[n]+\Omega_{I+1})[n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_{I+1}) \\ f_{\psi(\Omega_{I2}[n]+\Omega_{I+n})}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_{I+\omega}) \\ f_{\psi(\Omega_{I2}[n]+\Omega_{I+\psi_I(\Omega_{I+n})})}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_{I+\psi_I(\Omega_{I+\omega})}) \\ f_{\psi(\Omega_{I2})[n+1]}f_{\psi(\Omega_{I2})}(n)=f_{\psi(\Omega_{I2}[n]+\Omega_{I+\psi_I(\Omega_{I2})[n]})}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}+\Omega_{I+\psi_I(\Omega_{I2})}) \\ f_{\psi(\Omega_{I2})[2n]}f_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}2) \\ f^2_{\psi(\Omega_{I2})}(n) & & \psi(\Omega_{I2}\psi(\Omega_{I2})) \\ f_{\psi(\Omega_{I2})+1}(n) & & \psi(\Omega_{I2}\Omega) \\ f_{\psi(\Omega_{I2}\Omega)}(n) & & \psi(\Omega_{I2}\Omega_2) \\ f_{\psi(\Omega_{I2}\Omega_\omega)}(n) & & \psi(\Omega_{I2}\Omega_\omega) \end{eqnarray*}

Up to the limit of Catching function
=Catching Hierarchy I= Catching hierarchy I range from \(C(\Omega)\) to \(C(\alpha)\), where \(\alpha\) is a large 1-uncountable 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.