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Part 2 is here.

Now an ordinal function for SGH-catching-FGH-points comes in. -

## Catching function

I use C(α) to stand for this function. It's defined as follows:

1. Condition α=0: C(0) is first ordinal β such that g_β(n) is comparable to f_β(n);
2. Condition α=S: C(α+1) is the next ordinal β such that g_β(n) is comparable to f_β(n) after C(α);
3. Condition α=L: C(α)[n]=C(α[n]).

Also, C(α) is the smallest ordinal β such that g_β(n) is comparable to f_β(n) but it's larger than C(γ) for all γ<α.

So C(α) is the 1+α-th SGH-catching-FGH-point.

"Comparable" is an unclear term, but here we can say:

f_β(n) and g_β(n) are comparable iff exists k, for any n, g_β(n+k)>f_β(n).

For comparisons in part 2, we know C(0)=ψ(Ω_ω), C(1)=ψ(Ω_Ω_ω) and C(ω)=ψ(ψ_I(0)).

The last result in my part 2 shows that $$f_{\psi(\psi_I(0))}(n)\approx g_{\psi(\psi_I(0))}(n)$$, but what's $$f_{\psi(\psi_I(0))}(n+1)$$?

Remember what I've said in part 1 - ω's in SGH are orderless, so f_{ψ(ψ_I(0))}(n+1)≈g_{ψ(ψ_I(0))}(n+1)≈g_{ψ(Ω_Ω_...)}(n+1) with n+1 nests - or ω+1 nests. ψ_I(0) is the first fixed point that α->Ω_α, so ψ_I(0) means ω nests; and ψ_I(1) is the second fixed point that α->Ω_α, so ψ_I(1) means ω2 nests. So...

\begin{eqnarray*} FGH & & \text{SGH Ordinal} \\ \text{or Catching function} & & \text{normal ordinals} \\ f_{\psi(\psi_I(0))}(n+1) &~& \psi(\Omega_{\psi_I(0)+1}) \\ f_{\psi(\psi_I(0))}(n+2) &~& \psi(\Omega_{\Omega_{\psi_I(0)+1}}) \\ f_{\psi(\psi_I(0))}(2n) &~& \psi(\psi_I(1)) \\ f_{\psi(\psi_I(0))}(n^2) &~& \psi(\psi_I(\omega)) \\ f^2_{\psi(\psi_I(0))}(n) &~& \psi(\psi_I(\psi(\psi_I(0)))) \\ f_{\psi(\psi_I(0))+1}(n) &~& \psi(\psi_I(\Omega)) \\ f_{\psi(\psi_I(\Omega))}(n) &~& \psi(\psi_I(\Omega_2)) \\ f_{\psi(\psi_I(\Omega_\omega))}(n) &~& \psi(\psi_I(\Omega_\omega)) \\ C(\omega+1) &=& \psi(\psi_I(\Omega_\omega)) \\ C(\omega+2) &=& \psi(\psi_I(\Omega_{\Omega_\omega})) \\ C(\omega2) &=& \psi(\psi_I(\psi_I(0))) \\ C(\omega2+1) &=& \psi(\psi_I(\psi_I(\Omega_\omega))) \\ C(\omega3) &=& \psi(\psi_I(\psi_I(\psi_I(0)))) \\ C(\omega^2) &=& \psi(I) \\ f_{C(\omega^2)}(n+1) &~& \psi(I+\psi_I(I+\psi_I(I))) \\ f_{C(\omega^2)}(n+2) &~& \psi(I+\psi_I(I+\psi_I(I+\psi_I(I)))) \\ f_{C(\omega^2)}(2n) &~& \psi(I2) \end{eqnarray*}

To stand for ω^^(ω+1), we need ε(0)^ε(0), or ω^(ε(0)+1); to stand for ω^^(ω+2), we need ε(0)^ε(0)^ε(0), or ω^ω^(ε(0)+1) (though they're not equal, I don't care these between ω^^ω and ω^^(ω2). I just care what's ω^^(ω2)[n].); and ω^^(ω2)=ε(1).

In θ function, to stand for θ(θ(...θ(θ(0))...)) with ω+1 nests we need θ(θ(Ω),θ(Ω)+1); to stand for θ(θ(...θ(θ(0))...)) with ω+2 nests we need θ(θ(θ(Ω),θ(Ω)+1),θ(Ω)+1); and θ(θ(...θ(θ(0))...)) with ω2 nests is θ(Ω,1).

In ψ function, to stand for ψ(Ω^ψ(Ω^...ψ(Ω^ψ(Ω))...)) with ω+1 nests we need ψ(Ω^Ω+Ω^ψ(Ω^Ω)); to stand for ψ(Ω^ψ(Ω^...ψ(Ω^ψ(Ω))...)) with ω+2 nests we need ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^ψ(Ω^Ω))); and ψ(Ω^ψ(Ω^...ψ(Ω^ψ(Ω))...)) with ω2 nests is ψ((Ω^Ω)2).

If those make sense to you, you can continue reading my blog post.

\begin{eqnarray*} FGH & & \text{SGH Ordinal} \\ \text{or Catching function} & & \text{normal ordinals} \\ f_{C(\omega^2)+1}(n) &~& \psi(I\Omega) \\ f_{\psi(\psi_I(I\Omega))}(n) &~& \psi(I\Omega_2) \\ C(\omega^2+1) &=& \psi(I\Omega_\omega) \\ C(\omega^2+\omega) &=& \psi(I\psi_I(0)) \\ C(\omega^2+\omega+1) &=& \psi(I\psi_I(\Omega_\omega)) \\ C(\omega^2+\omega2) &=& \psi(I\psi_I(\psi_I(0))) \\ C(\omega^22) &=& \psi(I\psi_I(I)) \\ C(\omega^22+1) &=& \psi(I\psi_I(I\Omega_\omega)) \\ C(\omega^23) &=& \psi(I\psi_I(I\psi_I(I))) \\ C(\omega^3) &=& \psi(I^2) \\ C(\omega^3+1) &=& \psi(I^2\Omega_\omega) \\ C(\omega^32) &=& \psi(I^2\psi_I(I^2)) \\ C(\omega^4) &=& \psi(I^3) \\ C(\omega^5) &=& \psi(I^4) \\ C(\omega^\omega) &=& \psi(I^\omega) \\ f_{C(\omega^\omega)+1}(n) &~& \psi(I^\Omega) \\ C(\omega^\omega+1) &=& \psi(I^{\Omega_\omega}) \\ C(\omega^\omega+\omega) &=& \psi(I^{\psi_I(0)}) \\ C(\omega^\omega2) &=& \psi(I^{\psi_I(I^\omega)}) \\ C(\omega^{\omega+1}) &=& \psi(I^I) \\ f_{C(\omega^{\omega+1})}(n+1) &~& \psi(I^I+I^{\psi_I(I^I)}) \\ f_{C(\omega^{\omega+1})}(2n) &~& \psi(I^I2) \\ C(\omega^{\omega+1}+1) &=& \psi(I^I\Omega_\omega) \\ C(\omega^{\omega+1}2) &=& \psi(I^I\psi_I(I^I)) \\ C(\omega^{\omega+2}) &=& \psi(I^{I+1}) \\ C(\omega^{\omega+3}) &=& \psi(I^{I+2}) \\ C(\omega^{\omega2}) &=& \psi(I^{I+\omega}) \\ C(\omega^{\omega2}+1) &=& \psi(I^{I+\Omega_\omega}) \\ C(\omega^{\omega2}2) &=& \psi(I^{I+\psi_I(I^{I+\omega})}) \\ C(\omega^{\omega2+1}) &=& \psi(I^{I2}) \\ C(\omega^{\omega2+1}+1) &=& \psi(I^{I2}\Omega_\omega) \\ C(\omega^{\omega2+2}) &=& \psi(I^{I2+1}) \\ C(\omega^{\omega3}) &=& \psi(I^{I2+\omega}) \\ C(\omega^{\omega3+1}) &=& \psi(I^{I3}) \\ C(\omega^{\omega4+1}) &=& \psi(I^{I4}) \\ C(\omega^{\omega^2}) &=& \psi(I^{I\omega}) \\ C(\omega^{\omega^2+1}) &=& \psi(I^{I^2}) \\ C(\omega^{\omega^3}) &=& \psi(I^{I^2\omega}) \\ C(\omega^{\omega^\omega}) &=& \psi(I^{I^\omega}) \\ C(\omega^{\omega^{\omega2}}) &=& \psi(I^{I^{I+\omega}}) \\ C(\omega^{\omega^{\omega^\omega}}) &=& \psi(I^{I^{I^\omega}}) \\ C(\varepsilon_0) &=& \psi(\psi_{\Omega_{I+1}}(0)) \\ C(\varepsilon_0+1) &=& \psi(\psi_{\Omega_{I+1}}(\Omega_\omega)) \\ C(\varepsilon_0\omega) &=& \psi(\psi_{\Omega_{I+1}}(I)) \end{eqnarray*}

I'm not very sure about the comparisons beyond C(ε(0)ω). But, if you really want to know how strong the Catching function and BEAF are, you can make the comparisons yourself, since the C() is already defined well.

## BEAF Comparisons

But I can compare BEAF with catching function clearly. Notice the catching function can be used in both FGH and SGH.

\begin{eqnarray*} BEAF & & \text{ordinal Catching function} \\ \{L,1\}_{n,n}=\{n,n/2\} & & C(0) \\ \{L,2\}_{n,n}=\{n,n//2\} & & C(1) \\ \{L,3\}_{n,n}=\{n,n///2\} & & C(2) \\ \{L,X\}_{n,n}=\{n,n(1)/2\} & & C(\omega) \\ \{n,n/(1)/2\} & & C(\omega+1) \\ \{n,n//(1)/2\} & & C(\omega+2) \\ \{L,X+1\}_{n,n}=\{n,n(1)//2\} & & C(\omega2) \\ \{L,X+2\}_{n,n}=\{n,n(1)///2\} & & C(\omega3) \\ \{L,2X\}_{n,n}=\{n,n(1)(1)/2\} & & C(\omega^2) \\ \{n,n/(1)(1)/2\} & & C(\omega^2+1) \\ \{n,n(1)/(1)/2\} & & C(\omega^2+\omega) \\ \{L,2X+1\}_{n,n}=\{n,n(1)(1)//2\} & & C(\omega^22) \\ \{L,3X\}_{n,n}=\{n,n(1)(1)(1)/2\} & & C(\omega^3) \\ \{L,X^2\}_{n,n}=\{n,n(2)/2\} & & C(\omega^\omega) \\ \{L,X^2+1\}_{n,n}=\{n,n(2)//2\} & & C(\omega^\omega2) \\ \{L,X^2+X\}_{n,n}=\{n,n(2)(1)/2\} & & C(\omega^{\omega+1}) \\ \{L,X^22\}_{n,n}=\{n,n(2)(2)/2\} & & C(\omega^{\omega2}) \\ \{L,X^3\}_{n,n}=\{n,n(3)/2\} & & C(\omega^{\omega^2}) \\ \{L,X^4\}_{n,n}=\{n,n(4)/2\} & & C(\omega^{\omega^3}) \\ \{L,X^X\}_{n,n}=\{n,n(0,1)/2\} & & C(\omega^{\omega^\omega}) \\ \{L,X^{X^X}\}_{n,n}=\{n,n((1)1)/2\} & & C(\omega^{\omega^{\omega^\omega}}) \\ \{L,X\uparrow\uparrow X\}_{n,n} & & C(\varepsilon_0) \\ \{L,X\uparrow\uparrow(X+1)\}_{n,n} & & C(\varepsilon_0^{\varepsilon_0}) \\ \{L,X\uparrow\uparrow(2X)\}_{n,n} & & C(\varepsilon_1) \\ \{L,X\uparrow\uparrow X\uparrow\uparrow X\}_{n,n} & & C(\varepsilon_{\varepsilon_0}) \\ \{L,X\uparrow\uparrow\uparrow X\}_{n,n} & & C(\zeta_0) \\ \{L,\{X,X,4\}\}_{n,n} & & C(\varphi(3,0)) \\ \{L,\{X,X,X\}\}_{n,n} & & C(\varphi(\omega,0)) \\ \{L,\{X,2X,X\}\}_{n,n} & & C(\varphi(\omega,1)) \\ \{L,\{X,X,X+1\}\}_{n,n} & & C(\varphi(\omega+1,0)) \\ \{L,\{X,X,1,2\}\}_{n,n} & & C(\Gamma_0) \\ \{L,\{X,X,1,3\}\}_{n,n} & & C(\varphi(2,0,0)) \\ \{L,\{X,X,1,1,2\}\}_{n,n} & & C(\varphi(1,0,0,0)) \\ \{L,\{X,X(1)2\}\}_{n,n} & & C(\theta(\Omega^\omega)) \\ \{L,\{X,X+1(1)2\}\}_{n,n} & & C(\theta(\Omega^{\omega+1})) \\ \{L,\{X,X,2(1)2\}\}_{n,n} & & C(\theta(\Omega^\Omega)) \\ \{L,\{X,X,2(1)3\}\}_{n,n} & & C(\theta(\Omega^\Omega2)) \\ \{L,\{X,X,2(1)1,2\}\}_{n,n} & & C(\theta(\Omega^{\Omega+1})) \\ \{L,\{X,X,2(1)(1)2\}\}_{n,n} & & C(\theta(\Omega^{\Omega2})) \\ \{L,\{X,X,2(2)2\}\}_{n,n} & & C(\theta(\Omega^{\Omega^2})) \\ \{L,\{X,X,2(0,1)2\}\}_{n,n} & & C(\theta(\Omega^{\Omega^\Omega})) \\ \{L,\{X_2\uparrow\uparrow X_2\&X\}\}_{n,n} & & C(\theta(\varepsilon_{\Omega+1})) \\ \{L,\{X_3\&X_2\&X\}\}_{n,n} & & C(\theta(\Omega_2^\omega)) \\ \{L,2,2\}_{n,n}=\{L,L\}_{n,n} & & C(C(0)) \\ \{L,\{L,2\}\}_{n,n} & & C(C(1)) \\ \{L,\{L,X\}\}_{n,n} & & C(C(\omega)) \\ \{L,\{L,X^2\}\}_{n,n} & & C(C(\omega^\omega)) \\ \{L,\{L,X^X\}\}_{n,n} & & C(C(\omega^{\omega^\omega})) \\ \{L,\{L,X\uparrow\uparrow X\}\}_{n,n} & & C(C(\varepsilon_0)) \\ \{L,3,2\}_{n,n}=\{L,\{L,L\}\}_{n,n} & & C(C(C(0))) \\ \{L,4,2\}_{n,n} & & C(C(C(C(0)))) \\ \{L,X,2\}_{n,n} & & \alpha\mapsto C(\alpha) \end{eqnarray*}

So here, catching function itself is not enough for comparisons among BEAF, FGH and SGH. To continue, we need something more powerful base on Catching function. Then the Catching hierarchy comes in. -

## Catching hierarchy I

Now I use the first uncountable ordinal Ω for a diagonalizer in the C(). Imagine this: when we meet an Ω and want to solve it, first find the nearest C(), and copy things inside this C() but not including this Ω n times, and every time insert a copy to where the Ω is. In another word C(#Ω@)=C(#C(#...C(#0@)...@)@) with n nests, where thereis no ordinal in the "@". So

\begin{eqnarray*} BEAF & & \text{Catching function} \\ \{L,X,2\}_{n,n} & & C(\Omega) \\ \{L,\{L,X,2\}+1\}_{n,n} & & C(\Omega+C(\Omega)) \\ \{L,\{L,X,2\}+2\}_{n,n} & & C(\Omega+C(\Omega)2) \\ \{L,2\{L,X,2\}\}_{n,n} & & C(\Omega+C(\Omega)^2) \\ \{L,\{L,X,2\}^2\}_{n,n} & & C(\Omega+C(\Omega)^{C(\Omega)}) \\ \{L,\{L,\{L,X,2\}\}\}_{n,n} & & C(\Omega+C(\Omega+C(\Omega))) \\ \{L,2X,2\}_{n,n} & & C(\Omega2) \\ \{L,X^2,2\}_{n,n} & & C(\Omega\omega) \\ \{L,L,2\}_{n,n} & & C(\Omega C(0)) \\ \{L,\{L,X,2\},2\}_{n,n} & & C(\Omega C(\Omega)) \\ \{L,X,3\}_{n,n} & & C(\Omega^2) \\ \{L,2X,3\}_{n,n} & & C(\Omega^22) \\ \{L,X,4\}_{n,n} & & C(\Omega^3) \\ \{L,X,X\}_{n,n} & & C(\Omega^\omega) \\ \{L,X,L\}_{n,n} & & C(\Omega^{C(0)}) \\ \{L,X,\{L,X,2\}\}_{n,n} & & C(\Omega^{C(\Omega)}) \\ \{L,X,\{L,X,\{L,X,2\}\}\}_{n,n} & & C(\Omega^{C(\Omega^{C(\Omega)})}) \\ \{L,X,1,2\}_{n,n} & & C(\Omega^\Omega) \\ \{L,X,2,2\}_{n,n} & & C(\Omega^{\Omega+1}) \\ \{L,X,1,3\}_{n,n} & & C(\Omega^{\Omega2}) \\ \{L,X,1,1,2\}_{n,n} & & C(\Omega^{\Omega^2}) \\ \{L,X(1)2\}_{n,n} & & C(\Omega^{\Omega^\omega}) \end{eqnarray*}

Well, we come to a most confusing point - the breaking point of linear legiattic arrays, just as the comparisons between FGH and SGH!

\begin{eqnarray*} BEAF & & \text{Catching function} \\ \{L,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}) \\ \{L,\{L,2,1,...,1,2\}+1\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}+C(\Omega^{\Omega^\omega})) \\ \{L,\{L,2,1,...,1,2\},2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}+\Omega) \\ \{L,\{L,2,1,...,1,2\},1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}+\Omega^\Omega) \\ \{L,3,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}2) \\ \{L,X,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}\omega) \\ \{L,L,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega}C(0)) \\ \{L,X,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+1}) \\ \{L,2X,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+1}2) \\ \{L,X,3,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+2}) \\ \{L,X,L,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+C(0)}) \\ \{L,X,1,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega}) \\ \{L,X,2,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega+1}) \\ \{L,X,1,3,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega2}) \\ \{L,X,1,X,1,...,1,2\}_{n,n}\,2\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega\omega}) \\ \{L,X,1,1,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega^2}) \\ \{L,X,1,1,1,2,1,...,1,2\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega+\Omega^3}) \\ \{L,X,1,...,1,3\}_{n,n}\,3\,a.p\,X+1 & & C(\Omega^{\Omega^\omega2}) \\ \{L,X,1,...,1,4\}_{n,n}\,4\,a.p\,X+1 & & C(\Omega^{\Omega^\omega3}) \\ \{L,X,1,...,1,X\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega\omega}) \\ \{L,X+1(1)2\}_{n,n}=\{L,L,...,L,L\}_{n,n}\,a.p\,X+1 & & C(\Omega^{\Omega^\omega C(0)}) \\ \{L,X,1,...,1,1,2\}_{n,n}\,2\,a.p\,X+2 & & C(\Omega^{\Omega^{\omega+1}}) \\ \{L,X+2(1)2\}_{n,n}=\{L,L,...,L,L\}_{n,n}\,a.p\,X+2 & & C(\Omega^{\Omega^{\omega+1}C(0)}) \\ \{L,2X(1)2\}_{n,n} & & C(\Omega^{\Omega^{\omega2}}) \\ \{L,X^2(1)2\}_{n,n} & & C(\Omega^{\Omega^{\omega^2}}) \\ \{L,L(1)2\}_{n,n} & & C(\Omega^{\Omega^{C(0)}}) \\ \{L,X,2(1)2\}_{n,n} & & C(\Omega^{\Omega^\Omega}) \end{eqnarray*}

Though C() function is very strong and the Ω inside it seems to act strongly, changing "L,X" into "L,X,2" in the first row can still change ω into Ω in the C() function, just as the normal notation.

\begin{eqnarray*} BEAF & & \text{Catching function} \\ \{L,2X,2(1)2\}_{n,n} & & C(\Omega^{\Omega^\Omega}2) \\ \{L,L,2(1)2\}_{n,n} & & C(\Omega^{\Omega^\Omega}C(0)) \\ \{L,X,3(1)2\}_{n,n} & & C(\Omega^{\Omega^\Omega+1}) \\ \{L,X,1,2(1)2\}_{n,n} & & C(\Omega^{\Omega^\Omega+\Omega}) \\ \{L,X(1)3\}_{n,n} & & C(\Omega^{\Omega^\Omega+\Omega^\omega}) \\ \{L,X,2(1)3\}_{n,n} & & C(\Omega^{\Omega^\Omega2}) \\ \{L,X,2(1)4\}_{n,n} & & C(\Omega^{\Omega^\Omega3}) \\ \{L,X(1)L\}_{n,n} & & C(\Omega^{\Omega^\Omega C(0)}) \\ \{L,X(1)1,2\}_{n,n} & & C(\Omega^{\Omega^{\Omega+1}}) \\ \{L,X(1)1,3\}_{n,n} & & C(\Omega^{\Omega^{\Omega+1}2}) \\ \{L,X(1)1,1,2\}_{n,n} & & C(\Omega^{\Omega^{\Omega+2}}) \\ \{L,X(1)(1)2\}_{n,n} & & C(\Omega^{\Omega^{\Omega+\omega}}) \\ \{L,X,2(1)(1)2\}_{n,n} & & C(\Omega^{\Omega^{\Omega2}}) \\ \{L,X,2(1)(1)(1)2\}_{n,n} & & C(\Omega^{\Omega^{\Omega3}}) \\ \{L,X,2(2)2\}_{n,n} & & C(\Omega^{\Omega^{\Omega^2}}) \\ \{L,X,2(0,1)2\}_{n,n} & & C(\Omega^{\Omega^{\Omega^\Omega}}) \\ X_2\uparrow\uparrow X_2@n & & C(\varepsilon_{\Omega+1}) \end{eqnarray*}

We use X's in legiattic arrays, and X_2 means a line of X's. And we use X_3 in legiattic arrays on X_2, and X_4 means a line of X_3. etc. So we use things like ...X_6@X_4@X_2@n. If you understand the "nested structures" you can go on.

\begin{eqnarray*} BEAF & & \text{Catching function} \\ X_2\uparrow\uparrow\uparrow X_2@n & & C(\zeta_{\Omega+1}) \\ \{X_2,X_2,X_2\}@n & & C(\varphi(\omega,\Omega+1)) \\ \{X_2,X_2,1,2\}@n & & C(\Gamma_{\Omega+1}) \\ \{X_2,X_2(1)2\}@n & & C(\theta_1(\Omega_2^\omega)) \\ \{X_2,X_2,2(1)2\}@n & & C(\theta_1(\Omega_2^{\Omega_2})) \\ \{X_2,X_2,2(0,1)2\}@n & & C(\theta_1(\Omega_2^{\Omega_2^{\Omega_2}})) \\ X_3\uparrow\uparrow X_3\&X_2@n & & C(\theta_1(\theta_2(1))) \\ \{X_3,X_3,1,2\}\&X_2@n & & C(\theta_1(\Omega_3)) \\ X_4\&X_3\&X_2@n & & C(\theta_1(\Omega_3^\omega)) \\ \{X_2,X_2/2\}@n & & C(\psi_1(\Omega_\omega)) \\ \{X_2,X_2//2\}@n & & C(\psi_1(\Omega_{\Omega_\omega})) \\ \{X_2,X_2(1)/2\}@n & & C(\psi_1(\psi_I(0))) \\ \{X_2,X_2,2(1)/2\}@n & & C(\psi_1(\psi_I(\Omega_2))) \\ \{X_2,X_2/(1)/2\}@n & & C(\psi_1(\psi_I(\Omega_\omega))) \\ \{X_2,X_2(1)//2\}@n & & C(\psi_1(\psi_I(\psi_I(0)))) \\ \{X_2,X_2(1)(1)/2\}@n & & C(\psi_1(I)) \end{eqnarray*}

Catching hierarchy I is not enough for the comparisons now. To continue we need a higher catching function.

## Catching hierarchy II

We know, a catching ordinal must be something like ψ(α). That means, it's a fixed point that β->ω^β. And a cardinal α can be used as a "diagonalizer" in ψ_α(). In normal notation, $$\psi_{\Omega_{1+k}}()$$ also can be written as ψ_k() for positive integer k and ψ_Ω() also can be written as ψ(). Naturally a stronger catching hierarchy comes in.

• C_π(0)=ψ_π(Ω_ω)
• If C_π(α)=ψ_π(β), then C_π(α+1)=ψ_π(γ) where ψ(γ) is the least ordinal that $$g_{\psi(\gamma)}(n)$$ is comparable to $$f_{\psi(\gamma)}(n)$$ and γ>β and both ψ_π(β) and ψ_π(γ) are full-simplified.
• For limit α, C_π(α)[n]=C_π(α[n])
• π is diagonalizer of C_π() function.

C_{Ω_{1+k}}() also can be written as C_k() for positive integer k and C_Ω() also can be written as C().

- What's full-simplified?

- Notation ψ(β) is full-simplified iff ψ(β+1)>ψ(β). For example, ψ(Ω_2) is full-simplified but ψ(ψ_1(Ω_2)) isn't because ψ(Ω_2+1)>ψ(Ω_2)=ψ(ψ_1(Ω_2)+1)=ψ(ψ_1(Ω_2)). Sometimes the ψ function increases but sometimes it stays at a value, and a full-simplified notation is at an increasing part or at the end of a staying part.

That's it! This notation is much, MUCH stronger now. It's enough for comarisons up to (even beyond) the limit of BEAF.

\begin{eqnarray*} BEAF & & \text{Catching function} \\ \{L,X_3\}_{X_2,X_2}@n=\{X_2,X_2(1)/2\}@n & & C(C_1(\omega)) \\ \{L,X_3+1\}_{X_2,X_2}@n=\{X_2,X_2(1)//2\}@n & & C(C_1(\omega2)) \\ \{L,2X_3\}_{X_2,X_2}@n=\{X_2,X_2(1)(1)/2\}@n & & C(C_1(\omega^2)) \\ \{L,X_3^2\}_{X_2,X_2}@n=\{X_2,X_2(2)/2\}@n & & C(C_1(\omega^\omega)) \\ \{L,X_3^{X_3}\}_{X_2,X_2}@n=\{X_2,X_2(0,1)/2\}@n & & C(C_1(\omega^{\omega^\omega})) \\ \{L,X_3\uparrow\uparrow X_3\}_{X_2,X_2}@n & & C(C_1(\varepsilon_0)) \\ \{L,\{X_3,X_3,1,2\}\}_{X_2,X_2}@n & & C(C_1(\Gamma_0)) \\ \{L,L\}_{X_2,X_2}@n & & C(C_1(C(0))) \\ \{L,3,2\}_{X_2,X_2}@n & & C(C_1(C(C(0)))) \\ \{L,X_3,2\}_{X_2,X_2}@n & & C(C_1(C(\Omega))) \\ \{L,X_3,3\}_{X_2,X_2}@n & & C(C_1(C(\Omega^2))) \\ \{L,X_3,1,2\}_{X_2,X_2}@n & & C(C_1(C(\Omega^\Omega))) \\ \{L,X_3(1)2\}_{X_2,X_2}@n & & C(C_1(C(\Omega^{\Omega^\omega}))) \\ \{L,X_3,2(1)2\}_{X_2,X_2}@n & & C(C_1(C(\Omega^{\Omega^\Omega}))) \\ \{L,X_3,2(0,1)2\}_{X_2,X_2}@n & & C(C_1(C(\Omega^{\Omega^{\Omega^\Omega}}))) \\ X_4\uparrow\uparrow X_4@X_2@n & & C(C_1(C(\varepsilon_{\Omega+1}))) \\ X_5\&X_4@X_2@n & & C(C_1(C(\theta_1(\Omega_2^\omega)))) \\ \{X_4,X_4/2\}@X_2@n & & C(C_1(C(C_1(0)))) \\ \{L,X_5,2\}_{X_4,X_4}@X_2@n & & C(C_1(C(C_1(C(\Omega))))) \\ X_6@X_4@X_2@n & & C(C_1(C(C_1(C(\Omega^{\Omega^\omega}))))) \\ X_6\uparrow\uparrow X_6@X_4@X_2@n & & C(C_1(C(C_1(C(\varepsilon_{\Omega+1}))))) \\ \{L,X_7,2\}_{X_6,X_6}@X_4@X_2@n & & C(C_1(C(C_1(C(C_1(C(\Omega))))))) \\ \{n,n\backslash2\} & & C(C_1(\Omega)) \\ \{n,n\backslash\backslash2\} & & C(C_1(\Omega)+1) \\ \{n,n(1)\backslash2\} & & C(C_1(\Omega)+\omega) \\ \{L2,2X\}_{n,n} & & C(C_1(\Omega)+\omega^2) \\ \{L2,X^2\}_{n,n} & & C(C_1(\Omega)+\omega^\omega) \\ \{L2,X\uparrow\uparrow X\}_{n,n} & & C(C_1(\Omega)+\varepsilon_0) \\ \{L2,L\}_{n,n} & & C(C_1(\Omega)+C(0)) \\ \{L2,L2\}_{n,n} & & C(C_1(\Omega)+C(C_1(\Omega))) \\ \{L2,\{L2,2\}\}_{n,n} & & C(C_1(\Omega)+C(C_1(\Omega)+1)) \\ \{L2,\{L2,L2\}\}_{n,n} & & C(C_1(\Omega)+C(C_1(\Omega)+C(C_1(\Omega)))) \\ \{L2,X,2\}_{n,n} & & C(C_1(\Omega)+\Omega) \\ \{L2,X,1,2\}_{n,n} & & C(C_1(\Omega)+\Omega^\Omega) \\ \{L2,X,2(1)2\}_{n,n} & & C(C_1(\Omega)+\Omega^{\Omega^\Omega}) \\ \{L2,X,2(0,1)2\}_{n,n} & & C(C_1(\Omega)+\Omega^{\Omega^{\Omega^\Omega}}) \\ X_2\uparrow\uparrow X_2\%n & & C(C_1(\Omega)+\varepsilon_{\Omega+1}) \\ \{X_2,X_2/2\}\%n & & C(C_1(\Omega)+C_1(0)) \\ \{X_2,X_2\backslash2\}\%n & & C(C_1(\Omega)+C_1(C(C_1(\Omega)))) \\ \{L2,X_3,2\}_{X_2,X_2}\%n & & C(C_1(\Omega)+C_1(C(C_1(\Omega)+\Omega))) \\ \{X_4,X_4\backslash2\}\%X_2\%n & & C(C_1(\Omega)+C_1(C(C_1(\Omega)+C_1(C(C_1(\Omega)))))) \\ \{X_6,X_6\backslash2\}\%X_4\%X_2\%n & & C(C_1(\Omega)+C_1(C(C_1(\Omega)+ \\ & & C_1(C(C_1(\Omega)+C_1(C(C_1(\Omega)))))))) \\ \{n,n|2\} & & C(C_1(\Omega)2) \\ \{n,n-2\} & & C(C_1(\Omega)3) \\ \{LX\}_{n,n} & & C(C_1(\Omega)\omega) \end{eqnarray*}

To go further in BEAF clearly, we should use a more clear notation. just think of square-bracketed arrays like this:

{[L,A+1]}_b,p = b @& b @& ... b @& b - p times, where @& is "[L,A]-attic array of"

The rule 1 and 3 remain the same.

For example, in no-comma arrays we don't know what's {LL(1)LL,LL(1)LL}_n,n. It can be either {LL(1)LL,2,2}_n,n - a 3 entries LL(1)LL-attic array or a two-row array with its first row being LL(1)LL,LL and second row is LL. They're clear in square-bracketed arrays: {[L,L(1)L,L],[L,L(1)L,L]}_n,n and {[L,L(1)L,L],[L,L](1)[L,L]}_n,n.

Then [&] means "square-bracketed array of". Let's continue to the limit of BEAF.

\begin{eqnarray*} BEAF & & \text{Catching function} \\ \{[L,2X]\}_{n,n} & & C(C_1(\Omega)\omega2) \\ \{[L,X^X]\}_{n,n} & & C(C_1(\Omega)\omega^\omega) \\ \{[L,X_2\&X]\}_{n,n} & & C(C_1(\Omega)\theta(\Omega^\omega)) \\ \{[L,L]\}_{n,n} & & C(C_1(\Omega)C(0)) \\ \{[L,\{L,2\}]\}_{n,n} & & C(C_1(\Omega)C(1)) \\ \{[L,[L,2]]\}_{n,n} & & C(C_1(\Omega)C(C_1(\Omega))) \\ \{[L,[L,L]]\}_{n,n} & & C(C_1(\Omega)C(C_1(\Omega)C(0))) \\ \{[L,X,2]\}_{n,n} & & C(C_1(\Omega)\Omega) \\ \{[L,2X,2]\}_{n,n} & & C(C_1(\Omega)\Omega2) \\ \{[L,L,2]\}_{n,n} & & C(C_1(\Omega)\Omega C(0)) \\ \{[L,X,3]\}_{n,n} & & C(C_1(\Omega)\Omega^2) \\ \{[L,L,L]\}_{n,n} & & C(C_1(\Omega)\Omega^{C(0)}) \\ \{[L,X,1,2]\}_{n,n} & & C(C_1(\Omega)\Omega^\Omega) \\ \{[L,X,2,2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega+1}) \\ \{[L,X,1,3]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega2}) \\ \{[L,X,1,1,2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^2}) \\ \{[L,X(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^\omega}) \\ \{[L,[L,X(1)2]+1]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^\omega}+C_1(\Omega)) \\ \{[L,[L,X(1)2]+1,2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^\omega}+C_1(\Omega)\Omega) \\ \{[[L,X(1)2],X(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^\omega}2) \\ \{[L,4,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega}3) \\ \{[L,X,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega}\omega) \\ \{[L,X,2,1,...,1,2]\}_{n,n}\text{ a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+1}) \\ \{[L,X,3,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+2}) \\ \{[L,X,L,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+C(0)}) \\ \{[L,X,1,2,1,...,1,2]\}_{n,n}\text{ a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+\Omega}) \\ \{[L,X,1,3,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+\Omega2}) \\ \{[L,X,1,1,2,1,...,1,2]\}_{n,n}\text{ a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega+\Omega^2}) \\ \{[L,X,1,...,1,3]\}_{n,n}\text{ 3 a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega2}) \\ \{[L,X,1,...,1,X]\}_{n,n}\text{ a.p. X+1} & & C(C_1(\Omega)\Omega^{\Omega^\omega\omega}) \\ \{[L,X,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+2} & & C(C_1(\Omega)\Omega^{\Omega^{\omega+1}}) \\ \{[L,X,1,...,1,2]\}_{n,n}\text{ 2 a.p. X+3} & & C(C_1(\Omega)\Omega^{\Omega^{\omega+2}}) \\ \{[L,2X(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\omega2}}) \\ \{[L,X^2(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\omega^2}}) \\ \{[L,L(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{C(0)}}) \\ \{[L,X,2(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\Omega}}) \\ \{[L,X,2(1)(1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\Omega2}}) \\ \{[L,X,2(2)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\Omega^2}}) \\ \{[L,X,2(0,1)2]\}_{n,n} & & C(C_1(\Omega)\Omega^{\Omega^{\Omega^\Omega}}) \\ \{X_2\uparrow\uparrow X_2[\&]L\}_{n,n} & & C(C_1(\Omega)\varepsilon_{\Omega+1}) \\ \{X_3\&X_2[\&]L\}_{n,n} & & C(C_1(\Omega)\theta_1(\Omega_2^\omega)) \\ \{L[\&]L\}_{n,n} & & C(C_1(\Omega)C_1(0)) \\ \{[L,2][\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)))) \\ \{[L,L][\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)C(0)))) \\ \{[L,X_3,2][\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)\Omega))) \\ \{[L,X_3,1,2][\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)\Omega^\Omega))) \\ \{[L,X_3,2(1)2][\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)\Omega^{\Omega^\Omega}))) \\ \{X_4\uparrow\uparrow X_4[\&]L[\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)\varepsilon_{\Omega+1}))) \\ \{L[\&]L[\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)C_1(0)))) \\ \{[L,X_5,2][\&]L[\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)C_1(C(C_1(\Omega)\Omega))))) \\ \{[L,X_7,2][\&]L[\&]L[\&]L\}_{n,n} & & C(C_1(\Omega)C_1(C(C_1(\Omega)\times \\ & & C_1(C(C_1(\Omega)C_1(C(C_1(\Omega)\Omega))))))) \end{eqnarray*}

So the limit of BEAF is $$C(C_1(\Omega)^2)$$. And meameamealokkapoowa oompa is approximately $$f_{C(C_1(\Omega)C_1(C(C_1(\Omega)99+9))+9)}(10)$$. If you want to know the value in normal ordinal notations in FGH, just DO IT YOURSELF.

We can certainly go further in catching hierarchy: $$C(\Omega_2),C(\Omega_\omega),C(\Omega_\Omega),C(\Omega_{\Omega_\Omega}),...$$ get to $$C(\psi_I(0))$$, and $$C(C_I(0))=C(\psi_I(\Omega_\omega)),C(C_I(\omega^2))=C(\psi_I(I))$$. Further, we get $$C(I),C(\psi_{I_2}(0)),C(I_2),C(I(1,0))=C(\alpha\mapsto C_{I(1,0)}(\alpha)),C(\chi(M)),C(\chi(M_2)),C(\chi(M(1,0))),...$$ But, the χ function is higher a level function and only returns inaccessible ordinals.

## Catching hierarchy III

Think of the compact notation: $$\Psi_\pi(0,\alpha)$$ return "normal" ordinals, this means it works as the $$\psi_\pi(\alpha)$$ function. $$\Psi_\pi(1,\alpha)$$ return inaccessible ordinals, and it works as the $$\chi_\pi(\alpha)$$ function. In $$\Psi_\pi(\beta,\alpha)$$ the β can be any value, which means any level of the type of returns. If we apply these to the catching hierarchy, what will we get?

- Things like $$C_\Omega(0,M),C_\Omega(0,\Xi(3,0)),C_\Omega(0,\Xi(4,0)),C_\Omega(0,\Xi(K)),C_\Omega(0,\Xi(K^K)),C_\Omega(0,\varepsilon_{K+1})$$

$$,C_\Omega(0,K_2),C_\Omega(0,\alpha\mapsto K_\alpha),...$$ just think of this question:

We know the only difference between the normal notations (based on psi() function) and the catching hierarchy (based on C() function) is the definition of $$\psi_\pi(\alpha+1)$$ and $$C_\pi(\alpha+1)$$. But other things work the same. There're many ways to extend the notations both. But, does an ordinal exists such that catching hierarchy and normal notations use the same level notations (i.e. the only difference between them is symbol ψ and C, but cardinals are the same) ? If it exists, it marks the end of the catching hierarchy, and must be really, REALLY large!