10,825 Pages

Part 1 is here.

## Higher non-legion arrays

Something in BEAF, SGH and FGH are equivalent. (Here uncountable Ω's are in ordinal collasping functions.)

\begin{eqnarray*} BEAF &SGH& FGH \\ n &\omega& n \\ X &\Omega&  \omega \\ X_2 &\Omega_2&  \Omega \\ X_k &\Omega_k&  \Omega_{k-1} \end{eqnarray*}

Now I continue comparisons.

\begin{eqnarray*} BEAF & \text{FGH  Ordinal} & \text{SGH  Ordinal} \\ \{X,X,2(1)2\}\&n & \theta(\Omega^\Omega) & \theta(\Omega_2^{\Omega_2}) \\ \{X,X+1,2(1)2\}\&n & \theta(\Omega^{\theta(\Omega^\Omega)},\theta(\Omega^\Omega)+1) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\theta_1(\Omega_2^{\Omega_2})},\theta_1(\Omega_2^{\Omega_2})+1)) \\ \{X,2X,2(1)2\}\&n & \theta(\Omega^\Omega,1) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2},1)) \\ \{X,X^X,2(1)2\}\&n & \theta(\Omega^\Omega,\omega^\omega) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2},\Omega^\omega)) \\ \{X,3,3(1)2\}\&n & \theta(\Omega^\Omega,\theta(\Omega^\Omega)) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2},\theta_1(\Omega_2^{\Omega_2}))) \\ \{X,X,3(1)2\}\&n & \theta(\Omega^\Omega+1) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2}+1)) \\ \{X,X,4(1)2\}\&n & \theta(\Omega^\Omega+2) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2}+2)) \\ \{X,X,X(1)2\}\&n & \theta(\Omega^\Omega+\omega) & \theta(\Omega_2^{\Omega_2}+\theta_1(\Omega_2^{\Omega_2}+\omega)) \\ \{X,X,1,2(1)2\}\&n & \theta(\Omega^\Omega+\Omega) & \theta(\Omega_2^{\Omega_2}+\Omega_2) \\ \{X,X,1,1,2(1)2\}\&n & \theta(\Omega^\Omega+\Omega^2) & \theta(\Omega_2^{\Omega_2}+\Omega_2^2) \\ \{X,X(1)3\}\&n & \theta(\Omega^\Omega+\Omega^\omega) & \theta(\Omega_2^{\Omega_2}+\Omega_2^\omega) \\ \{X,X,2(1)3\}\&n & \theta(\Omega^\Omega2) & \theta(\Omega_2^{\Omega_2}2) \\ \{X,X,2(1)4\}\&n & \theta(\Omega^\Omega3) & \theta(\Omega_2^{\Omega_2}3) \\ \{X,X(1)X\}\&n & \theta(\Omega^\Omega\omega) & \theta(\Omega_2^{\Omega_2}\omega) \\ \{X,X(1)1,2\}\&n & \theta(\Omega^{\Omega+1}) & \theta(\Omega_2^{\Omega_2+1}) \\ \{X,X,2(1)2,2\}\&n & \theta(\Omega^{\Omega+1}+\Omega^\Omega) & \theta(\Omega_2^{\Omega_2+1}+\Omega_2^{\Omega_2}) \\ \{X,X(1)1,3\}\&n & \theta(\Omega^{\Omega+1}2) & \theta(\Omega_2^{\Omega_2+1}2) \\ \{X,X(1)X,X\}\&n & \theta(\Omega^{\Omega+1}\omega) & \theta(\Omega_2^{\Omega_2+1}\omega) \\ \{X,X(1)1,1,2\}\&n & \theta(\Omega^{\Omega+2}) & \theta(\Omega_2^{\Omega_2+2}) \\ \{X,X(1)1,1,1,2\}\&n & \theta(\Omega^{\Omega+3}) & \theta(\Omega_2^{\Omega_2+3}) \\ \{X,X(1)(1)2\}\&n & \theta(\Omega^{\Omega+\omega}) & \theta(\Omega_2^{\Omega_2+\omega}) \\ \{X,X,2(1)(1)2\}\&n & \theta(\Omega^{\Omega2}) & \theta(\Omega_2^{\Omega_22}) \\ \{X,X,2(1)(1)(1)2\}\&n & \theta(\Omega^{\Omega3}) & \theta(\Omega_2^{\Omega_23}) \\ \{X,X(2)2\}\&n & \theta(\Omega^{\Omega\omega}) & \theta(\Omega_2^{\Omega_2\omega}) \\ \{X,X,2(2)2\}\&n & \theta(\Omega^{\Omega^2}) & \theta(\Omega_2^{\Omega_2^2}) \\ \{X,X,2(2)(2)2\}\&n & \theta(\Omega^{\Omega^22}) & \theta(\Omega_2^{\Omega_2^22}) \\ \{X,X,2(3)2\}\&n & \theta(\Omega^{\Omega^3}) & \theta(\Omega_2^{\Omega_2^3}) \\ \{X,X,2(0,1)2\}\&n & \theta(\Omega^{\Omega^\Omega}) & \theta(\Omega_2^{\Omega_2^{\Omega_2}}) \\ \{X,X,2((1)1)2\}\&n & \theta(\Omega^{\Omega^{\Omega^\Omega}}) & \theta(\Omega_2^{\Omega_2^{\Omega_2^{\Omega_2}}}) \\ X_2\uparrow\uparrow X_2\&X\&n & \theta(\theta_1(1)) & \theta(\theta_2(1)) \\ \{X,X,2(X_2\uparrow\uparrow X_2)2\}\&n & \theta(\theta_1(1,\Omega)) & \theta(\theta_2(1,\Omega_2)) \\ X_2\uparrow\uparrow(2X_2)\&X\&n & \theta(\theta_1(1,\Omega+1)) & \theta(\theta_2(1,\Omega_2+1)) \\ X_2\uparrow\uparrow\uparrow X_2\&X\&n & \theta(\theta_1(2)) & \theta(\theta_2(2)) \\ \{X_2,X_2,X_2\}\&X\&n & \theta(\theta_1(\omega)) & \theta(\theta_2(\omega)) \\ \{X_2,X_2,1,2\}\&X\&n & \theta(\Omega_2) & \theta(\Omega_3) \\ \{X_2,X_2,1,3\}\&X\&n & \theta(\Omega_22) & \theta(\Omega_32) \\ \{X_2,X_2,1,1,2\}\&X\&n & \theta(\Omega_2^2) & \theta(\Omega_3^2) \\ \{X_2,X_2(1)2\}\&X\&n & \theta(\Omega_2^\omega) & \theta(\Omega_3^\omega) \\ \{X_2,X_2,2(1)2\}\&X\&n & \theta(\Omega_2^{\Omega_2}) & \theta(\Omega_3^{\Omega_3}) \\ \{X_2,X_2,2(1)3\}\&X\&n & \theta(\Omega_2^{\Omega_2}2) & \theta(\Omega_3^{\Omega_32}) \\ \{X_2,X_2,2(1)1,2\}\&X\&n & \theta(\Omega_2^{\Omega_2+1}) & \theta(\Omega_3^{\Omega_3+1}) \\ \{X_2,X_2,2(1)(1)2\}\&X\&n & \theta(\Omega_2^{\Omega_22}) & \theta(\Omega_3^{\Omega_32}) \\ \{X_2,X_2,2(2)2\}\&X\&n & \theta(\Omega_2^{\Omega_2^2}) & \theta(\Omega_3^{\Omega_3^2}) \\ \{X_2,X_2,2(0,1)2\}\&X\&n & \theta(\Omega_2^{\Omega_2^{\Omega_2}}) & \theta(\Omega_3^{\Omega_3^{\Omega_3}}) \\ X_3\uparrow\uparrow X_3\&X_2\&X\&n & \theta(\theta_2(1)) & \theta(\theta_3(1)) \\ X_3\uparrow\uparrow\uparrow X_3\&X_2\&X\&n & \theta(\theta_2(2)) & \theta(\theta_3(2)) \\ \{X_3,X_3,1,2\}\&X_2\&X\&n & \theta(\Omega_3) & \theta(\Omega_4) \\ X_4\&X_3\&X_2\&X\&n & \theta(\Omega_3^\omega) & \theta(\Omega_4^\omega) \\ X_5\&X_4\&X_3\&X_2\&X\&n & \theta(\Omega_4^\omega) & \theta(\Omega_5^\omega) \end{eqnarray*}

So the limit of a legion is $$\theta(\Omega_\omega)$$, which is also the limit of BAN, which also "happens to be" the first SGH-catching-up-FGH point.

## Next legion

Do you remember this: "When SGH grows a part of its ordinal from 2,3,4,... to ω (it's new, called "active point"), and at the same time FGH grows its ordinal to a limit one, then FGH ordinal increasing by 1 will change the SGH active point (the new ω) into Ω in ordinal collasping functions"?

So we can compare SGH with FGH from "inside" with this property. We don't run into difficulty to compare SGH with FGH and BEAF from "outside". Instead, we should make the active point clear.

\begin{eqnarray*} BEAF & FGH & \text{SGH Ordinal} \\ \{n,n/2\} & f_{\theta(\Omega_\omega)}(n) & \theta(\Omega_\omega) \\ \{n,n+1/2\} & f_{\theta(\Omega_\omega)}(n+1) & \theta(\Omega_{\omega+1}) \\ \{n,2n/2\} & f_{\theta(\Omega_\omega)}(2n) & \theta(\Omega_{\omega2}) \\ \{n,n^n/2\} & f_{\theta(\Omega_\omega)}(n^n) & \theta(\Omega_{\omega^\omega}) \\ \{n,3,2/2\} & f^2_{\theta(\Omega_\omega)}(n) & \theta(\Omega_{\theta(\Omega_\omega)}) \\ \{n,n,2/2\} & f_{\theta(\Omega_\omega)+1}(n) & \theta(\Omega_\Omega) \\ \{n,n+1,2/2\} & f_{\theta(\Omega_\omega)+1}(n+1) & \theta(\Omega_{\theta(\Omega_\Omega)},\theta(\Omega_\Omega)+1) \\ \{n,2n,2/2\} & f_{\theta(\Omega_\omega)+1}(2n) & \theta(\Omega_\Omega,1) \\ \{n,3,3/2\} & f^2_{\theta(\Omega_\omega)+1}(n) & \theta(\Omega_\Omega,\theta(\Omega_\Omega)) \\ \{n,n,3/2\} & f_{\theta(\Omega_\omega)+2}(n) & \theta(\Omega_\Omega+1) \\ \{n,n,4/2\} & f_{\theta(\Omega_\omega)+3}(n) & \theta(\Omega_\Omega+2) \\ \{n,n,1,2/2\} & f_{\theta(\Omega_\omega)+\omega+1}(n) & \theta(\Omega_\Omega+\Omega) \\ \{n,n,2,2/2\} & f_{\theta(\Omega_\omega)+\omega+2}(n) & \theta(\Omega_\Omega+\Omega+1) \\ \{n,n,1,3/2\} & f_{\theta(\Omega_\omega)+\omega2+1}(n) & \theta(\Omega_\Omega+\Omega2) \\ \{n,n,1,1,2/2\} & f_{\theta(\Omega_\omega)+\omega^2+1}(n) & \theta(\Omega_\Omega+\Omega^2) \\ \{n,n(1)2/2\} & f_{\theta(\Omega_\omega)+\omega^\omega}(n) & \theta(\Omega_\Omega+\Omega^\omega) \\ \{n,n(1)3/2\} & f_{\theta(\Omega_\omega)+\omega^\omega2}(n) & \theta(\Omega_\Omega+\Omega^\Omega+\Omega^\omega) \\ \{n,n(1)1,2/2\} & f_{\theta(\Omega_\omega)+\omega^{\omega+1}}(n) & \theta(\Omega_\Omega+\Omega^{\Omega+1}) \\ \{n,n(1)(1)2/2\} & f_{\theta(\Omega_\omega)+\omega^{\omega2}}(n) & \theta(\Omega_\Omega+\Omega^{\Omega+\omega}) \\ \{n,n(2)2/2\} & f_{\theta(\Omega_\omega)+\omega^{\omega^2}}(n) & \theta(\Omega_\Omega+\Omega^{\Omega\omega}) \\ \{n,n(0,1)2/2\} & f_{\theta(\Omega_\omega)+\omega^{\omega^\omega}}(n) & \theta(\Omega_\Omega+\Omega^{\Omega^\omega}) \\ \{X\uparrow\uparrow X\&n/2\} & f_{\theta(\Omega_\omega)+\varepsilon_0}(n) & \theta(\Omega_\Omega+\varepsilon_{\Omega+1}) \\ \{\{X,X,1,2\}\&n/2\} & f_{\theta(\Omega_\omega)+\Gamma_0}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_2)) \\ \{\{X,X(1)2\}\&n/2\} & f_{\theta(\Omega_\omega)+\theta(\Omega^\omega)}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_2^\omega)) \\ \{X_2\uparrow\uparrow X_2\&X\&n/2\} & f_{\theta(\Omega_\omega)+\theta(\varepsilon_{\Omega+1})}(n) & \theta(\Omega_\Omega+\theta_1(\varepsilon_{\Omega_2+1})) \\ \{X_3\&X_2\&X\&n/2\} & f_{\theta(\Omega_\omega)+\theta(\Omega_2^\omega)}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_3^\omega)) \\ \{n,n/3\} & f_{\theta(\Omega_\omega)2}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\omega)) \\ \{n,n,2/3\} & f_{\theta(\Omega_\omega)2+1}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)) \\ \{n,n,2/4\} & f_{\theta(\Omega_\omega)3+1}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)2) \\ \{n,n/n\} & f_{\theta(\Omega_\omega)\omega}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\omega) \\ \{n,n/1,2\} & f_{\theta(\Omega_\omega)\omega+1}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\Omega) \\ \{n,n/1,3\} & f_{\theta(\Omega_\omega)\omega2+1}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\Omega2) \\ \{n,n/1,1,2\} & f_{\theta(\Omega_\omega)\omega^2+1}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\Omega^2) \\ \{n,n/1(1)2\} & f_{\theta(\Omega_\omega)\omega^\omega}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\Omega^\omega) \\ \{n,n/X_2\&X\&n\} & f_{\theta(\Omega_\omega)\theta(\Omega^\omega)}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\theta_1(\Omega_2^\omega)) \\ \{n,n/1/2\}=2\&\&n & f_{\theta(\Omega_\omega)^2}(n) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)\theta_1(\Omega_\omega)) \end{eqnarray*}

Now you see how big the big boowa, great big boowa, grand boowa, super gongulus and the wompogulus are!

## Linear legion arrays and higher

1&&n is {n,n/2} and has growth rate θ(Ω_ω) in FGH. 2&&n is {n,n/1/2} and has growth rate θ(Ω_ω)^2. In this part you'll see: An "&&n" maps SGH into FGH approximately too.

\begin{eqnarray*} BEAF & \text{FGH Ordinal} & \text{SGH Ordinal} \\ \{n,n,2/1/2\} & \theta(\Omega_\omega)^2+1 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^2) \\ \{n,n/2/2\} & \theta(\Omega_\omega)^2+\theta(\Omega_\omega) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^2+\theta_1(\Omega_\omega)) \\ \{n,n/1/3\} & \theta(\Omega_\omega)^22 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^2+\theta_1(\Omega_\Omega)\theta_1(\Omega_\omega)) \\ \{n,n/1/1/2\}=3\&\&n & \theta(\Omega_\omega)^3 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^2\theta_1(\Omega_\omega)) \\ \{n,n/1/1/1/2\}=4\&\&n & \theta(\Omega_\omega)^4 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^3\theta_1(\Omega_\omega)) \\ \{n,n(/1)2\}=X\&\&n & \theta(\Omega_\omega)^\omega & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^\omega) \\ \{n,n,2(/1)2\} & \theta(\Omega_\omega)^\omega+1 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^\Omega) \\ \{n,n(/1)3\} & \theta(\Omega_\omega)^\omega2 & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^\Omega+\theta_1(\Omega_\Omega)^\omega) \\ \{n,n(/1)1/2\}=X+1\&\&n & \theta(\Omega_\omega)^{\omega+1} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^\Omega\theta_1(\Omega_\omega)) \\ \{n,n(/1)(/1)2\}=2X\&\&n & \theta(\Omega_\omega)^{\omega2} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\Omega+\omega}) \\ \{n,n(/2)2\}=X^2\&\&n & \theta(\Omega_\omega)^{\omega^2} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\Omega\omega}) \\ X\uparrow\uparrow X\&\&n & \theta(\Omega_\omega)^{\varepsilon_0} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\varepsilon_{\Omega+1}}) \\ \{X,X/2\}\&\&n & \theta(\Omega_\omega)^{\theta(\Omega_\omega)} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\theta_1(\Omega_\omega)}) \\ \{X,X/2\}+1\&\&n & \theta(\Omega_\omega)^{\theta(\Omega_\omega)+1} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\theta_1(\Omega_\Omega)}\theta_1(\Omega_\omega)) \\ \{X,X/2\}^{\{X,X/2\}}\&\&n & \theta(\Omega_\omega)^{\theta(\Omega_\omega)^{\theta(\Omega_\omega)}} & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega)^{\theta_1(\Omega_\Omega)^{\theta_1(\Omega_\omega)}}) \\ \{X,X/2\}\uparrow\uparrow X\&\&n & \varepsilon_{\theta(\Omega_\omega)+1} & \theta(\Omega_\Omega+\varepsilon_{\theta_1(\Omega_\Omega)+1}) \\ \text{I don't care here} & \theta(\Omega^\omega,\theta(\Omega_\omega)+1) & \theta(\Omega_\Omega+\theta_1(\Omega_2^\omega,\theta_1(\Omega_\Omega)+1)) \\ \text{I don't care} & \theta(\Omega_2^\omega,\theta(\Omega_\omega)+1) & \theta(\Omega_\Omega+\theta_1(\Omega_3^\omega,\theta_1(\Omega_\Omega)+1)) \\ \text{I don't care} & \theta(\Omega_\omega,1) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega,1)) \\ \text{don't care} & \theta(\Omega_\omega,\theta(\Omega_\omega)) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega,\theta_1(\Omega_\omega))) \\ \text{don't care} & \theta(\Omega_\omega+1) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega+1)) \\ \text{don't care} & \theta(\Omega_\omega+\theta(\Omega_\omega)) & \theta(\Omega_\Omega+\theta_1(\Omega_\Omega+\theta_1(\Omega_\omega))) \\ \text{don't care} & \theta(\Omega_\omega+\Omega) & \theta(\Omega_\Omega+\Omega_2) \\ \text{don't care} & \theta(\Omega_\omega+\Omega_2) & \theta(\Omega_\Omega+\Omega_3) \\ \text{don't care here} & \theta(\Omega_\omega2) & \theta(\Omega_\Omega+\Omega_\omega) \\ \text{don't care here} & \theta(\Omega_\omega^{\Omega_\omega}) & \theta(\Omega_\Omega^{\Omega_\omega}) \\ \text{I don't care here} & \theta(\varepsilon_{\Omega_\omega+1}) & \theta(\varepsilon_{\Omega_\Omega+1}) \\ \{X,X+1/2\}\&\&n & \theta(\Omega_{\omega+1}) & \theta(\Omega_{\Omega+1}) \\ \{X,2X/2\}\&\&n & \theta(\Omega_{\omega2}) & \theta(\Omega_{\Omega+\omega}) \\ \{X,X^2/2\}\&\&n & \theta(\Omega_{\omega^2}) & \theta(\Omega_{\Omega\omega}) \\ \{X,3,2/2\}\&\&n & \theta(\Omega_{\theta(\Omega_\omega)}) & \theta(\Omega_{\theta_1(\Omega_\omega)}) \\ \{X,X,2/2\}\&\&n & \theta(\Omega_\Omega) & \theta(\Omega_{\Omega_2}) \\ \{X_2,X_2/2\}\&\&X\&\&n & \theta(\Omega_\Omega)^{\theta(\Omega_\omega)} & \theta(\Omega_{\Omega_2}+\theta_1(\Omega_{\Omega_2})^{\theta_1(\Omega_\omega)}) \\ \{X_2,X_2,2/2\}\&\&X\&\&n & \theta(\Omega_{\Omega_2}) & \theta(\Omega_{\Omega_3}) \\ \{X_3,X_3,2/2\}\&\&X_2\&\&X\&\&n & \theta(\Omega_{\Omega_3}) & \theta(\Omega_{\Omega_4}) \end{eqnarray*}

ps. Why can I say {X,X,2/2}&&n has growth rate θ(Ω_Ω) in FGH? Because {n,n,2/2} has growth rate θ(Ω_Ω) in SGH and the &&n maps SGH into FGH!

The limit of a legion legion is $$\theta(\Omega_{\Omega_\omega})$$, which is the second SGH-catching-FGH-point.

But it's not just by chance. Think like this: With the same growth rate, FGH ordinal is not more than SGH ordinal. In "&n"-chains, when I add an "&n", SGH ordinal will jump a step and FGH will follow a step to where SGH used to be. They meet at the limit of & operator, which is a legion. Similarly, in "&&n"-chains, when I add an "&&n", SGH ordinal will jump a step and FGH will follow a step to where SGH used to be. They meet at the limit of && operator, which is a legion legion.

Now you see how big the guapamonga and guapamongaplex are!

## Multiple legion arrays

Actually, adding an "&n", "&&n", "&&&n", "&(1)&n" or "A(&)n" where A(&) means an array of & will all maps SGH into FGH. So it's easy to compare them now. Their limit, /, //, ///, /(1)/ or A(/) are all SGH-catching-FGH-points.

\begin{eqnarray*} BEAF & \text{FGH Ordinal} & \text{SGH Ordinal} \\ \{n,n//2\} & \theta(\Omega_{\Omega_\omega}) & \theta(\Omega_{\Omega_\omega}) \\ \{n,n,2//2\} & \theta(\Omega_{\Omega_\omega})+1 & \theta(\Omega_{\Omega_\Omega}) \\ \{n,n,3//2\} & \theta(\Omega_{\Omega_\omega})+2 & \theta(\Omega_{\Omega_\Omega}+1) \\ \{n,n//3\} & \theta(\Omega_{\Omega_\omega})2 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\omega})) \\ \{n,n,2//4\} & \theta(\Omega_{\Omega_\omega})3+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})2) \\ \{n,n//1,2\} & \theta(\Omega_{\Omega_\omega})\omega+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})\Omega) \\ \{n,n//1//2\} & \theta(\Omega_{\Omega_\omega})^2 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})\theta_1(\Omega_{\Omega_\omega})) \\ \{n,n,2//1//2\} & \theta(\Omega_{\Omega_\omega})^2+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^2) \\ \{n,n,2//1//1//2\} & \theta(\Omega_{\Omega_\omega})^3+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^3) \\ \{n,n,2(//1)2\} & \theta(\Omega_{\Omega_\omega})^\omega+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^\Omega) \\ \{n,n,2(//2)2\} & \theta(\Omega_{\Omega_\omega})^{\omega^2}+1 & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^{\Omega^2}) \\ X\uparrow\uparrow X\&\&\&n & \theta(\Omega_{\Omega_\omega})^{\varepsilon_0} & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^{\varepsilon_{\Omega+1}}) \\ \{X,X//2\}\&\&\&n & \theta(\Omega_{\Omega_\omega})^{\theta(\Omega_{\Omega_\omega})} & \theta(\Omega_{\Omega_\Omega}+\theta_1(\Omega_{\Omega_\Omega})^{\theta_1(\Omega_{\Omega_\omega})}) \\ \{X,X//2\}\uparrow\uparrow X\&\&\&n & \varepsilon_{\theta(\Omega_{\Omega_\omega})+1} & \theta(\Omega_{\Omega_\Omega}+\varepsilon_{\theta_1(\Omega_{\Omega_\Omega})+1}) \\ \{X,X,2//2\}\&\&\&n & \theta(\Omega_{\Omega_\Omega}) & \theta(\Omega_{\Omega_{\Omega_2}}) \\ \{X_2,X_2,2//2\}\&\&\&X\&\&\&n & \theta(\Omega_{\Omega_{\Omega_2}}) & \theta(\Omega_{\Omega_{\Omega_3}}) \\ \{n,n///2\} & \theta(\Omega_{\Omega_{\Omega_\omega}}) & \theta(\Omega_{\Omega_{\Omega_\omega}}) \\ \{n,n,2///2\} & \theta(\Omega_{\Omega_{\Omega_\omega}})+1 & \theta(\Omega_{\Omega_{\Omega_\Omega}}) \\ \{X,X///2\}\&\&\&\&n & \theta(\Omega_{\Omega_{\Omega_\omega}})^{\theta(\Omega_{\Omega_{\Omega_\omega}})} & \theta(\Omega_{\Omega_{\Omega_\Omega}}+\theta_1(\Omega_{\Omega_{\Omega_\Omega}})^{\theta_1(\Omega_{\Omega_{\Omega_\omega}})}) \\ \{X,X,2///2\}\&\&\&\&n & \theta(\Omega_{\Omega_{\Omega_\Omega}}) & \theta(\Omega_{\Omega_{\Omega_{\Omega_2}}}) \\ \{n,n////2\} & \theta(\Omega_{\Omega_{\Omega_{\Omega_\omega}}}) & \theta(\Omega_{\Omega_{\Omega_{\Omega_\omega}}}) \\ \{n,n/////2\} & \theta(\Omega_{\Omega_{\Omega_{\Omega_{\Omega_\omega}}}}) & \theta(\Omega_{\Omega_{\Omega_{\Omega_{\Omega_\omega}}}}) \\ \{n,n(1)/2\} & \psi(\psi_I(0)) & \psi(\psi_I(0)) \end{eqnarray*}

See the relation between BEAF and SGH-catching-FGH-points?

Okey, the limit of a legion is the first SGH-catching-FGH-point, the limit of a legion legion is the second SGH-catching-FGH-point and the limit of a legion legion legion is the 3rd SGH-catching-FGH-point. Generally {n,n//...//2} (with k /'s) reach the k-th SGH-catching-FGH-point. Now you see how big the big hoss, grand hoss and great big hoss are!

To continue the comparisons, I have to use an ordinal function -- catching function. (Part 3)