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