FANDOM


Do you think the first time SGH catches up FGH at LVO or \(\psi(\Omega_\omega)\) ?

(See here.)

Do you think the limit of a legion of BEAF is LVO now?

We used to think the limit of a legion of BEAF is LVO, which "happens to be" the first catching ordinal some people think. Now we know the real catching ordinal, so let's analysis BEAF again. I hope to see the real strength of BEAF. Maybe it's stronger than BAN.

Let's go.

Linear arrays

In SGH, the n in \(g_\alpha(n)\) never change because \(g_{\alpha+1}(n)=g_\alpha(n)+1\) and \(g_\alpha(n)=g_{\alpha[n]}(n)\) if a limit ordinal. Then the ω's in SGH mean always n and they can change to n freely. And ω's are "orderless" in SGH. In FGH and HH we cannot do that because the n is changing.

BEAF FGH SGH Ordinal
{n,n,2} \(f_3(n)\) \(\varepsilon_0\)
{n,n+1,2} \(f_3(n+1)\approx f_2f_3(n)\) \(\varepsilon_0^\omega\)
{n,2n,2} \(f_3(2n)\approx f_2^nf_3(n)\) \(\varepsilon_1\)
{n,{n,n,2},2}={n,3,3} \(f_3^2(n)\) \(\varepsilon_{\varepsilon_0}\)
{n,n,3} \(f_4(n)\) \(\zeta_0\)
{n,n+1,3} \(f_4(n+1)\approx f_3f_4(n)\) \(\varepsilon_{\zeta_0+1}\)
{n,2n,3} \(f_4(2n)\approx f_3^nf_4(n)\) \(\zeta_{\zeta_0}\)
{n,n,4} \(f_5(n)\) \(\varphi(3,0)\)
{n,n,n} \(f_\omega(n)\) \(\varphi(\omega,0)\)
{n,n,n+1} \(f_\omega(n+1)\)

\(\varphi(\omega+1,0)\)

(omega+1 can change into 1+omega

and n+1 because it's orderless)

{n,n,2n} \(f_\omega(2n)\) \(\varphi(\omega2,0)\)
{n,n,{n,n,n}}={n,3,1,2} \(f_\omega^2(n)\) \(\varphi(\varphi(\omega,0),0)\)
{n,n,1,2} \(f_{\omega+1}(n)\) \(\Gamma_0\)
{n,n+1,1,2} \(f_{\omega+1}(n+1)\approx f_\omega f_{\omega+1}(n)\) \(\varphi(\Gamma_0,\Gamma_0+1)\)
{n,2n,1,2} \(f_{\omega+1}(2n)\approx f_\omega^nf_{\omega+1}(n)\) \(\Gamma_1\)
{n,3,2,2} \(f^2_{\omega+1}(n)\) \(\Gamma_{\Gamma_0}\)
{n,n,2,2} \(f_{\omega+2}(n)\) \(\varphi(1,1,0)\)
{n,n,3,2} \(f_{\omega+3}(n)\) \(\varphi(1,2,0)\)
{n,n,n,2} \(f_{\omega2}(n)\) \(\varphi(1,\omega,0)\)
{n,n,2n,2} \(f_{\omega2}(2n)\) \(\varphi(1,\omega2,0)\)
{n,3,1,3} \(f^2_{\omega2}(n)\) \(\varphi(1,\varphi(1,\omega,0),0)\)
{n,n,1,3} \(f_{\omega2+1}(n)\) \(\varphi(2,0,0)\)
{n,n,2,3} \(f_{\omega2+2}(n)\) \(\varphi(2,1,0)\)
{n,n,1,4} \(f_{\omega3+1}(n)\) \(\varphi(3,0,0)\)
{n,n,n,n} \(f_{\omega^2}(n)\) \(\varphi(\omega,0,0)\)
{n,n,1,1,2} \(f_{\omega^2+1}(n)\) \(\varphi(1,0,0,0)\)
{n,n,2,1,2} \(f_{\omega^2+2}(n)\) \(\varphi(1,0,1,0)\)
{n,n,1,2,2} \(f_{\omega^2+\omega+1}(n)\) \(\varphi(1,1,0,0)\)
{n,n,1,1,3} \(f_{\omega^22+1}(n)\) \(\varphi(2,0,0,0)\)
{n,n,1,1,1,2} \(f_{\omega^3+1}(n)\) \(\varphi(1,0,0,0,0)\)
{n,n,1,1,1,1,2} \(f_{\omega^4+1}(n)\) \(\varphi(1,0,0,0,0,0)\)
{n,n(1)2}={n,2,1,...1,2} the 2 is at position n+1 \(f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega)\)
{n,3,1,...1,2} 2 a.p n+1 \(f_{\omega^n}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega,1)\)
{n,n,1,...1,2} 2 a.p n+1 \(f^n_{\omega^n}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega,\omega)\)
{n,3,2,...1,2} a.p n+1 \(f_{\omega^n+1}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega,\theta(\Omega^\omega))\)
{n,n,2,...1,2} a.p n+1 \(f^n_{\omega^n+1}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+1)\)
{n,n,3,...1,2} 2 a.p n+1 \(f^n_{\omega^n+2}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+2)\)
{n,n,n,1,...1,2} 2 a.p n+1 \(f_{\omega^n+n}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\omega)\)
{n,n,1,2,1,...1,2} a.p n+1 \(f^n_{\omega^n+\omega}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega)\)
{n,n,n,2,1,...1,2} a.p n+1 \(f_{\omega^n+\omega+n}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega+\omega)\)
{n,n,1,3,1,...1,2} 2 a.p n+1 \(f^n_{\omega^n+\omega2}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega2)\)
{n,n,1,n,1,...1,2} 2 a.p n+1 \(f_{\omega^n+\omega n}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega\omega)\)
{n,n,1,1,2,1,...1,2} a.p n+1 \(f^n_{\omega^n+\omega^2}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega^2)\)
{n,n,1,1,1,2,1,...1,2} a.p n+1 \(f^n_{\omega^n+\omega^3}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega+\Omega^3)\)
{n,n,1,...1,3} 3 a.p n+1 \(f_{\omega^n2}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega2)\)
{n,n,1,...1,4} 4 a.p n+1 \(f_{\omega^n3}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega3)\)
{n,n+1(1)2}={n,n,n,...n} a.p n+1 \(f_{\omega^\omega}(n+1)\approx f_{\omega^nn}f_{\omega^\omega}(n)\) \(\theta(\Omega^\omega\omega)\)
{n,n,1,...1,2} 2 a.p n+2 \(f^n_{\omega^{n+1}}f_{\omega^\omega}(n)\) \(\theta(\Omega^{\omega+1})\)
{n,n+2(1)2}={n,n,n,...n} a.p n+2 \(f_{\omega^\omega}(n+2)\approx f_{\omega^{n+1}n}f_{\omega^\omega}(n)\) \(\theta(\Omega^{\omega+1}\omega)\)
{n,n,1,...1,2} 2 a.p n+3 \(f^n_{\omega^{n+2}}f_{\omega^\omega}(n)\) \(\theta(\Omega^{\omega+2})\)
{n,2n(1)2} \(f_{\omega^\omega}(2n)\approx f_{\omega^{2n}}f_{\omega^\omega}(n)\) \(\theta(\Omega^{\omega2})\)
{n,{n,n,2}(1)2} \(f_{\omega^\omega}f_3(n)\approx f_{\omega^{f_3(n)}}f_{\omega^\omega}(n)\) \(\theta(\Omega^{\varepsilon_0})\)
{n,3,2(1)2} \(f^2_{\omega^\omega}(n)\) \(\theta(\Omega^{\theta(\Omega^\omega)})\)
{n,4,2(1)2} \(f^3_{\omega^\omega}(n)\) \(\theta(\Omega^{\theta(\Omega^{\theta(\Omega^\omega)})})\)
{n,n,2(1)2} \(f_{\omega^\omega+1}(n)\) \(\theta(\Omega^\Omega)\)

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.

From dimensional to tetrational arrays

Question: What's the active point of θ(Ω^Ω) ? I can't find any ω's.

Answer: An Ω in ordinal collasping function means ω nests, or n nests in SGH. Then θ(Ω^θ(Ω^Ω),θ(Ω^Ω)+1) means ω+1 nests (or n+1 nests in SGH), and θ(Ω^Ω,1) means ω2 nests (or 2n nests in SGH).

BEAF FGH SGH Ordinal
{n,2n,2(1)2} \(f_{\omega^\omega+1}(2n)\) \(\theta(\Omega^\Omega,1)\)
{n,n,3(1)2} \(f_{\omega^\omega+2}(n)\) \(\theta(\Omega^\Omega+1)\)
{n,n,4(1)2} \(f_{\omega^\omega+3}(n)\) \(\theta(\Omega^\Omega+2)\)
{n,n,n(1)2} \(f_{\omega^\omega+\omega}(n)\) \(\theta(\Omega^\Omega+\omega)\)
{n,n,1,2(1)2} \(f_{\omega^\omega+\omega+1}(n)\) \(\theta(\Omega^\Omega+\Omega)\)
{n,n,1,3(1)2} \(f_{\omega^\omega+\omega2+1}(n)\) \(\theta(\Omega^\Omega+\Omega2)\)
{n,n,1,1,2(1)2} \(f_{\omega^\omega+\omega^2+1}(n)\) \(\theta(\Omega^\Omega+\Omega^2)\)
{n,n(1)3} \(f_{\omega^\omega2}(n)\) \(\theta(\Omega^\Omega+\Omega^\omega)\)
{n,n,2(1)3} \(f_{\omega^\omega2+1}(n)\) \(\theta(\Omega^\Omega2)\)
{n,n,2(1)4} \(f_{\omega^\omega3+1}(n)\) \(\theta(\Omega^\Omega3)\)
{n,n(1)n} \(f_{\omega^{\omega+1}}(n)\) \(\theta(\Omega^\Omega\omega)\)
{n,n(1)1,2} \(f_{\omega^{\omega+1}+1}(n)\) \(\theta(\Omega^{\Omega+1})\)
{n,n(1)1,3} \(f_{\omega^{\omega+1}2+1}(n)\) \(\theta(\Omega^{\Omega+1}2)\)
{n,n(1)1,1,2} \(f_{\omega^{\omega+2}+1}(n)\) \(\theta(\Omega^{\Omega+2})\)
{n,n(1)1,1,1,2} \(f_{\omega^{\omega+3}+1}(n)\) \(\theta(\Omega^{\Omega+3})\)
{n,n(1)(1)2} \(f_{\omega^{\omega2}}(n)\) \(\theta(\Omega^{\Omega+\omega})\)
{n,n,2(1)(1)2} \(f_{\omega^{\omega2}+1}(n)\) \(\theta(\Omega^{\Omega2})\)
{n,n,2(1)(1)(1)2} \(f_{\omega^{\omega3}+1}(n)\) \(\theta(\Omega^{\Omega3})\)
{n,n(2)2}=X^2&n \(f_{\omega^{\omega^2}}(n)\) \(\theta(\Omega^{\Omega\omega})\)
{n,n,2(2)2} \(f_{\omega^{\omega^2}+1}(n)\) \(\theta(\Omega^{\Omega^2})\)
{n,n(3)2}=X^3&n \(f_{\omega^{\omega^3}}(n)\) \(\theta(\Omega^{\Omega^2\omega})\)
{n,n(0,1)2}=X^X&n \(f_{\omega^{\omega^\omega}}(n)\) \(\theta(\Omega^{\Omega^\omega})\)
{n,n,2(0,1)2} \(f_{\omega^{\omega^\omega}+1}(n)\) \(\theta(\Omega^{\Omega^\Omega})\)
{n,n,2(0,1)3} \(f_{\omega^{\omega^\omega}2+1}(n)\) \(\theta(\Omega^{\Omega^\Omega}2)\)
{n,n,2(0,1)(0,1)2} \(f_{\omega^{\omega^\omega2}+1}(n)\) \(\theta(\Omega^{\Omega^\Omega2})\)
{n,n,2(1,1)2} \(f_{\omega^{\omega^{\omega+1}}+1}(n)\) \(\theta(\Omega^{\Omega^{\Omega+1}})\)
{n,n,2(0,2)2} \(f_{\omega^{\omega^{\omega2}}+1}(n)\) \(\theta(\Omega^{\Omega^{\Omega2}})\)
{n,n,2(0,0,1)2} \(f_{\omega^{\omega^{\omega^2}}+1}(n)\) \(\theta(\Omega^{\Omega^{\Omega^2}})\)
{n,n,2((1)1)2} \(f_{\omega^{\omega^{\omega^\omega}}+1}(n)\) \(\theta(\Omega^{\Omega^{\Omega^\Omega}})\)
{n,n((0,1)1)2}=X^X^X^X&n \(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(n)\) \(\theta(\Omega^{\Omega^{\Omega^{\Omega^\omega}}})\)
X^^X&n \(f_{\varepsilon_0}(n)\) \(\theta(\varepsilon_{\Omega+1})=\theta(\theta_1(1))\)
{n,n,2(X^^X)2} \(f_{\varepsilon_0+1}(n)\) \(\theta(\varepsilon_{\Omega2})=\theta(\theta_1(1,\Omega))\)
{n,n,2(X^^X)3} \(f_{\varepsilon_02+1}(n)\) \(\theta(\varepsilon_{\Omega2}2)\)
{n,n,2(X^^X)(X^^X)2} \(f_{\varepsilon_0^2+1}(n)\) \(\theta(\varepsilon_{\Omega2}^2)\)
{n,n,2(X^^X*X)2} \(f_{\varepsilon_0^\omega+1}(n)\) \(\theta(\varepsilon_{\Omega2}^\Omega)\)
{n,n,2((X^^X)^2)2} \(f_{\varepsilon_0^{\varepsilon_0}+1}(n)\) \(\theta(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}})\)
{n,n,2((X^^X)^X)2} \(f_{\varepsilon_0^{\varepsilon_0^\omega}+1}(n)\) \(\theta(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^\Omega})\)
(X^^X)^(X^^X)&n approx. X^^(X+1)&n \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}(n)\) \(\theta(\varepsilon_{\Omega2}^{\varepsilon_{\Omega2}^{\varepsilon_{\Omega+1}}})\)
X^^(2X)&n \(f_{\varepsilon_1}(n)\) \(\theta(\varepsilon_{\Omega2+1})\)
{n,n,2(X^^(2X))2} \(f_{\varepsilon_1+1}(n)\) \(\theta(\varepsilon_{\Omega3})\)
X^^(3X)&n \(f_{\varepsilon_2}(n)\) \(\theta(\varepsilon_{\Omega3+1})\)
X^^(X^2)&n \(f_{\varepsilon_\omega}(n)\) \(\theta(\varepsilon_{\Omega\omega})\)
X^^(X^3)&n \(f_{\varepsilon_{\omega^2}}(n)\) \(\theta(\varepsilon_{\Omega^2\omega})\)
X^^(X^X)&n \(f_{\varepsilon_{\omega^\omega}}(n)\) \(\theta(\varepsilon_{\Omega^\omega})\)
X^^X^^3&n \(f_{\varepsilon_{\omega^{\omega^\omega}}}(n)\) \(\theta(\varepsilon_{\Omega^{\Omega^\omega}})\)
X^^X^^X&n=X^^^3&n \(f_{\varepsilon_{\varepsilon_0}}(n)\) \(\theta(\varepsilon_{\varepsilon_{\Omega+1}})\)

Amazingly, what an "&n" does is --- Mapping SGH into FGH of the same ordinal approximately!

That means, if an array function p(n) has growth rate α in SGH, p(X)&n will have growth rate α in FGH approximately (more accurate: ω^α in FGH). And what a normal array does to SGH will be what an (X-array)&n does to FGH.

Linear array of arrays

BEAF FGH Ordinal SGH Ordinal
X^^^X&n \(\zeta_0\) \(\theta(\zeta_{\Omega+1})\)
X^^^(2X)&n \(\zeta_1\) \(\theta(\zeta_{\Omega2+1})\)
X^^^^3&n \(\zeta_{\zeta_0}\) \(\theta(\zeta_{\zeta_{\Omega+1}})\)
X^^^^X&n \(\varphi(3,0)\) \(\theta(\varphi(3,\Omega+1))=\theta(\theta_1(3))\)
{X,X,5}&n \(\varphi(4,0)\) \(\theta(\theta_1(4))\)
{X,X,X}&n \(\varphi(\omega,0)\) \(\theta(\theta_1(\omega))\)
{X,X,X}^{X,X,X}&n \(\varphi(\omega,0)^{\varphi(\omega,0)^{\varphi(\omega,0)}}\) \(\theta(\theta_1(\Omega)^{\theta_1(\Omega)^{\theta_1(\omega)}})\)
{X,X,X}^^X&n \(\varepsilon_{\varphi(\omega,0)+1}\) \(\theta(\theta(1,\theta_1(\Omega)+1))\)
{X,X,X}^^^X&n \(\zeta_{\varphi(\omega,0)+1}\) \(\theta(\theta(2,\theta_1(\Omega)+1))\)
{{X,X,X},X,X}&n approx. {X,2X,X}&n \(\varphi(\omega,1)\) \(\theta(\theta(\omega,\theta_1(\Omega)+1))\)
{n,n,2({X,2X,X})2} \(\varphi(\omega,1)+1\) \(\theta(\theta_1(\Omega,1))\)
{n,n,2({X,3X,X})2} \(\varphi(\omega,2)+1\) \(\theta(\theta_1(\Omega,2))\)
{n,n,2({X,{X,X,X},X})2} \(\varphi(\omega,\varphi(\omega,0))+1\) \(\theta(\theta_1(\Omega,\theta_1(\Omega)))\)
{X,X,X+1}&n \(\varphi(\omega+1,0)\) \(\theta(\theta_1(\Omega+1))\)
{X,X,X+2}&n \(\varphi(\omega+2,0)\) \(\theta(\theta_1(\Omega+2))\)
{X,X,2X}&n \(\varphi(\omega2,0)\) \(\theta(\theta_1(\Omega+\omega))\)
{X,X,X^2}&n \(\varphi(\omega^2,0)\) \(\theta(\theta_1(\Omega\omega))\)
{X,X,X^X}&n \(\varphi(\omega^\omega,0)\) \(\theta(\theta_1(\Omega^\omega))\)
{X,3,1,2}&n \(\varphi(\varphi(\omega,0),0)\) \(\theta(\theta_1(\theta_1(\omega)))\)
{X,X,1,2}&n \(\Gamma_0\) \(\theta(\theta_1(\Omega_2))=\theta(\Omega_2)\)
{X,2X,1,2}&n \(\Gamma_1\) \(\theta(\Omega_2+\theta_1(\Omega_2,1))\)
{X,X^X,1,2}&n \(\Gamma_{\omega^\omega}\) \(\theta(\Omega_2+\theta_1(\Omega_2,\Omega^\omega))\)
{X,3,2,2}&n \(\Gamma_{\Gamma_0}\) \(\theta(\Omega_2+\theta_1(\Omega_2,\theta_1(\Omega_2)))\)
{X,X,2,2}&n \(\varphi(1,1,0)\) \(\theta(\Omega_2+\theta_1(\Omega_2+1))\)
{X,3,3,2}&n \(\varphi(1,1,\varphi(1,1,0))\) \(\theta(\Omega_2+\theta_1(\Omega_2+1,\theta_1(\Omega_2+1)))\)
{X,X,3,2}&n \(\varphi(1,2,0)\) \(\theta(\Omega_2+\theta_1(\Omega_2+2))\)
{X,X,X,2}&n \(\varphi(1,\omega,0)\) \(\theta(\Omega_2+\theta_1(\Omega_2+\omega))\)
{X,3,1,3}&n \(\varphi(1,\varphi(1,\omega,0),0)\) \(\theta(\Omega_2+\theta_1(\Omega_2+\theta_1(\Omega_2+\omega)))\)
{X,X,1,3}&n \(\varphi(2,0,0)\) \(\theta(\Omega_22)\)
{X,X,2,3}&n \(\varphi(2,1,0)\) \(\theta(\Omega_22+\theta_1(\Omega_22+1))\)
{X,X,1,4}&n \(\varphi(3,0,0)\) \(\theta(\Omega_23)\)
{X,X,X,X}&n \(\varphi(\omega,0,0)\) \(\theta(\Omega_2\omega)\)
{X,X,1,1,2}&n \(\varphi(1,0,0,0)\) \(\theta(\Omega_2^2)\)
{X,X,1,1,3}&n \(\varphi(2,0,0,0)\) \(\theta(\Omega_2^22)\)
{X,X,1,1,1,2}&n \(\varphi(1,0,0,0,0)\) \(\theta(\Omega_2^3)\)
{X,X,1,1,1,1,2}&n \(\varphi(1,0,0,0,0,0)\) \(\theta(\Omega_2^4)\)
{X,X(1)2}&n={X,X,...,X}&n with X X's ={X,2,1,...1,2}&n a.p X+1 \(\theta(\Omega^\omega)\) \(\theta(\Omega_2^\omega)\)
{X,X(1)2}^^X&n \(\varepsilon_{\theta(\Omega^\omega)+1}\) \(\theta(\Omega_2^\Omega+\theta_1(1,\theta_1(\Omega_2^\Omega)+1))\)
{{X,X(1)2},X,1,2}&n \(\Gamma_{\theta(\Omega^\omega)+1}\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2,\theta_1(\Omega_2^\Omega)+1))\)
{{X,X(1)2},X(1)2}&n={X,3,1,...1,2}&n 2 a.p X+1 \(\theta(\Omega^\omega,1)\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2^\Omega,1))\)
{X,4,1,...1,2}&n 2 a.p X+1 \(\theta(\Omega^\omega,2)\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2^\Omega,2))\)
{X,3,2,1,...1,2}&n a.p X+1 \(\theta(\Omega^\omega,\theta(\Omega^\omega))\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2^\Omega,\theta_1(\Omega_2^\omega)))\)
{X,X,2,1,...1,2}&n a.p X+1 \(\theta(\Omega^\omega+1)\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2^\Omega+1))\)
{X,X,3,1,...1,2}&n 2 a.p X+1 \(\theta(\Omega^\omega+2)\) \(\theta(\Omega_2^\Omega+\theta_1(\Omega_2^\Omega+2))\)
{X,X,1,2,1,...1,2}&n a.p X+1 \(\theta(\Omega^\omega+\Omega)\) \(\theta(\Omega_2^\Omega+\Omega_2)\)
{X,X,1,1,2,1,...1,2}&n a.p X+1 \(\theta(\Omega^\omega+\Omega^2)\) \(\theta(\Omega_2^\Omega+\Omega_2^2)\)
{X,X,1,...1,3}&n 3 a.p X+1 \(\theta(\Omega^\omega2)\) \(\theta(\Omega_2^\Omega+\Omega_2^\omega))\)
{X,X,1,...1,4}&n 4 a.p X+1 \(\theta(\Omega^\omega3)\) \(\theta(\Omega_2^\Omega2+\Omega_2^\omega))\)
{X,X+1(1)2}&n={X,X,...X,X}&n a.p X+1 \(\theta(\Omega^\omega\omega)\) \(\theta(\Omega_2^\Omega\omega))\)
{X,3,1,...1,2}&n 2 a.p X+2 \(\theta(\Omega^\omega\theta(\Omega^\omega\omega))\) \(\theta(\Omega_2^\Omega\theta_1(\Omega_2^\Omega\omega)))\)
{X,X,1,...1,2}&n 2 a.p X+2 \(\theta(\Omega^{\omega+1})\) \(\theta(\Omega_2^{\Omega+1}))\)
{X,X,1,...1,3}&n 3 a.p X+2 \(\theta(\Omega^{\omega+1}2)\) \(\theta(\Omega_2^{\Omega+1}2))\)
{X,X+2(1)2}&n \(\theta(\Omega^{\omega+1}\omega)\) \(\theta(\Omega_2^{\Omega+1}\omega))\)
{X,X,1,...1,2}&n 2 a.p X+3 \(\theta(\Omega^{\omega+2})\) \(\theta(\Omega_2^{\Omega+2}))\)
{X,X+3(1)2}&n \(\theta(\Omega^{\omega+2}\omega)\) \(\theta(\Omega_2^{\Omega+2}\omega))\)
{X,2X(1)2}&n \(\theta(\Omega^{\omega2})\) \(\theta(\Omega_2^{\Omega+\omega}))\)
{X,3X(1)2}&n \(\theta(\Omega^{\omega3})\) \(\theta(\Omega_2^{\Omega2+\omega}))\)
{X,X^2(1)2}&n \(\theta(\Omega^{\omega^2})\) \(\theta(\Omega_2^{\Omega\omega}))\)
{X,X^X(1)2}&n \(\theta(\Omega^{\omega^\omega})\) \(\theta(\Omega_2^{\Omega^\omega}))\)
{X,X^^X(1)2}&n \(\theta(\Omega^{\varepsilon_0})\) \(\theta(\Omega_2^{\varepsilon_{\Omega+1}}))\)
{X,3,2(1)2}&n \(\theta(\Omega^{\theta(\Omega^\omega)})\) \(\theta(\Omega_2^{\theta_1(\Omega_2^\omega)}))\)
{X,4,2(1)2}&n \(\theta(\Omega^{\theta(\Omega^{\theta(\Omega^\omega)})})\) \(\theta(\Omega_2^{\theta_1(\Omega_2^{\theta_1(\Omega_2^\omega)})}))\)
{X,X,2(1)2}&n \(\theta(\Omega^\Omega)\) \(\theta(\Omega_2^{\Omega_2})\)

Now you see, we get LVO in FGH here, not a legion.

In Bowers's page he uses things like "b&b&...b&b - p times", but the & operator has 2 properties:- One is "sequence" - the & strings don't output a number, and they catch this structure into the outside arrays. e.g. {3&3(1)3&3}={3,3,3(1)3,3,3}, and it's not {tritri(1)tritri}=tritri. The other is "holdleft" - things lie left of & can't be solve before we solve the &. e.g. triakulus=3&3&3 but it's not tritri&3={3,tritri(1)2}={3,3,2(1)2}. The second property make it hard to solve this arrays, so we have to use some symbols. Let X represent a line of n's in normal arrays, X2 represent a line of X's in X-arrays, and Xk+1 represent a line of Xk's in Xk-arrays. Now "b&b&...b&b - p times" become \(X_{p-1}\&X_{p-2}\&...X_2\&X\&b\). You see, the & string is "layer-by-layer", not linear. Don't misunderstand it to X&X&...X&b !

So {X,n+1(1)2}&n is approximately {X,X(1)2}&(n+1), which is much less than {X,X+1(1)2}&n. And X2+1&X&n is far above them because it's {X,X,...X(1)X}&n - X X's in the first row. By the way, in "{X,X,1,...1,2} 2 a.p X+2", position X+2 doesn't mean "the 2nd entry in the 2nd row in the X-array" - it's just the X+2-th entry in the first row in the X-array. Notice that X only means a line of n in normal arrays, not a line of everything including X in X-arrays. To represent a line of X, we should use X2. Also, using Bowers's original notation and the "clear" notation, triakulus=3&3&3=X2&X&3, golapulus=10^100&10&10=X2^100&X&10, golapulusplex=10^100&10&10&10=X3^100&X2&X&10.

Does it make sense now?

Okey. Next I'll continue the comparisons. Part 2

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