FANDOM


BEAF isn't so strong as you think, it reaches only \(\theta(\Omega^{\Omega}2)\).

Things below \(\lbrace L,L\rbrace_{n,n}\) can be seen here.Now we begin with \(\lbrace L,L\rbrace_{n,n}\).

\(\lbrace L,L\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})2}\)

\(\lbrace L,L*n\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})\omega}\)

\(\lbrace L,L^2\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})^2}\)

\(\lbrace L,L^n\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})^{\omega}}\)

\(\lbrace L,L^L\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})^{\theta(\Omega^{\Omega})}}\)

\(\lbrace L,4,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\theta(\Omega^{\Omega})^{\theta(\Omega^{\Omega})^{\theta(\Omega^{\Omega})}}}\)

\(\lbrace L,X,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\theta(\Omega^{\Omega})+1}}\)

\(\lbrace L,X*2,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\theta(\Omega^{\Omega})+2}}\)

\(\lbrace L,X^2,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\theta(\Omega^{\Omega})+\omega}}\)

\(\lbrace L,X^X,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\theta(\Omega^{\Omega})+\omega^{\omega}}}\)

\(\lbrace L,L,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\theta(\Omega^{\Omega})2}}\)

\(\lbrace L, \lbrace L,L,2\rbrace ,2\rbrace_{n,n}\)has growth rate \(\epsilon_{\epsilon_{\epsilon_{\theta(\Omega^{\Omega})2}}}\)

\(\lbrace L,X,3\rbrace_{n,n}\)has growth rate \(\zeta_{\theta(\Omega^{\Omega})+1}\)

\(\lbrace L,X,4\rbrace_{n,n}\)has growth rate \(\varphi(3,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,X\rbrace_{n,n}\)has growth rate \(\varphi(\omega,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,L\rbrace_{n,n}\)has growth rate \(\varphi(\theta(\Omega^{\Omega}),\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,2\rbrace_{n,n}\)has growth rate \(\Gamma_{\theta(\Omega^{\Omega})+1}\)

\(\lbrace L,X*2,1,2\rbrace_{n,n}\)has growth rate \(\Gamma_{\theta(\Omega^{\Omega})+2}\)

\(\lbrace L,X,2,2\rbrace_{n,n}\)has growth rate \(\varphi(1,1,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,3,2\rbrace_{n,n}\)has growth rate \(\varphi(1,2,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,3\rbrace_{n,n}\)has growth rate \(\varphi(2,0,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,4\rbrace_{n,n}\)has growth rate \(\varphi(3,0,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,1,2\rbrace_{n,n}\)has growth rate \(\varphi(1,0,0,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,1,1,2\rbrace_{n,n}\)has growth rate \(\varphi(1,0,0,0,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X*2(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega},\theta(\Omega^{\Omega})+2)\)

\(\lbrace L,X,2(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}+1,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,2(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}+\Omega,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X,1,1,2(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}+\Omega^2,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)3\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}2,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)X\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}\omega,\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)L\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega}\theta(\Omega^{\Omega}),\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)1,2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega+1},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)1,1,2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega+2},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega2},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(1)(1)(1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega3},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(2)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega^2},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(3)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega^3},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X(0,1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega^{\omega}},\theta(\Omega^{\Omega})+1)\)

\(\lbrace L,X((1)1)2\rbrace_{n,n}\)has growth rate \(\theta(\Omega^{\omega^2},\theta(\Omega^{\Omega})+1)\)

n^^n@n has growth rate \(\theta(\Omega^{\epsilon_0},\theta(\Omega^{\Omega})+1)\)

{n,n,1,2}@n has growth rate \(\theta(\Omega^{\Gamma_0},\theta(\Omega^{\Omega})+1)\)

{n,n/2}@n has growth rate \(\theta(\Omega^{\theta(\Omega^{\Omega})},\theta(\Omega^{\Omega})+1)\)

n@n@n has growth rate \(\theta(\Omega^{\theta(\Omega^\omega,\theta(\Omega^{\Omega})+1)},\theta(\Omega^{\Omega})+1)\)

n@n@n@n has growth rate \(\theta(\Omega^{\theta(\Omega^{\theta(\Omega^\omega,\theta(\Omega^{\Omega})+1)},\theta(\Omega^{\Omega})+1)},\theta(\Omega^{\Omega})+1)\)

{n,n \ 2} has growth rate \(\theta(\Omega^{\Omega},1)\)

{n,n | 2} has growth rate \(\theta(\Omega^{\Omega},2)\)

{n,n - 2} has growth rate \(\theta(\Omega^{\Omega},3)\)

\(\lbrace LX \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega},\omega)\)

\(\lbrace LL \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega},\theta(\Omega^{\Omega}))\)

In order to make the notation clear, I use square-bracketed arrays (e.g. [L,L,L,X] ) instead of no-comma arrays (e.g. LLLX).The base is always L. The initial rule of these arrays become:

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

We can continue...

\(\lbrace [L,[L,L]] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega},\theta(\Omega^{\Omega},\theta(\Omega^{\Omega})))\)

\(\lbrace [L,X,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+1)\)

\(\lbrace [L,X*2,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+1,1)\)

\(\lbrace [L,L,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+1,\theta(\Omega^{\Omega}))\)

\(\lbrace [L,X,3] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+2)\)

\(\lbrace [L,X,4] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+3)\)

\(\lbrace [L,X,L] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\theta(\Omega^{\Omega}))\)

\(\lbrace [L,X,1,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega)\)

\(\lbrace [L,X,2,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega+1)\)

\(\lbrace [L,X,1,3] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega2)\)

\(\lbrace [L,X,1,L] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega\theta(\Omega^{\Omega}))\)

\(\lbrace [L,X,1,1,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^2)\)

\(\lbrace [L,X,1,1,1,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^3)\)

\(\lbrace [L,X(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega})\)

\(\lbrace [L,X*2(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega},1)\)

\(\lbrace [L,X,2(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega}+1)\)

\(\lbrace [L,X,1,2(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega}+\Omega)\)

\(\lbrace [L,X,1,1,2(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega}+\Omega^2)\)

\(\lbrace [L,X(1)3] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega}2)\)

\(\lbrace [L,X(1)L] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega}\theta(\Omega^{\Omega}))\)

\(\lbrace [L,X(1)1,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega+1})\)

\(\lbrace [L,X(1)1,1,2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega+2})\)

\(\lbrace [L,X(1)(1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega2})\)

\(\lbrace [L,X(2)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega^2})\)

\(\lbrace [L,X(0,1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega^{\omega}})\)

\(\lbrace [L,X((1)1)2] \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\omega^{\omega^{\omega}}})\)

Now I use [&] to represent "square-bracketed array of", then

\(\lbrace X\uparrow\uparrow X[\&]L \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\epsilon_0})\)

\(\lbrace L[\&]L \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\theta(\Omega^{\Omega})})\)

\(\lbrace [L,2][\&]L \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\theta(\Omega^{\Omega},1)})\)

\(\lbrace L[\&]L[\&]L \rbrace_{n,n}\) has growth rate \(\theta(\Omega^{\Omega}+\Omega^{\theta(\Omega^{\Omega}+\Omega^{\theta(\Omega^{\Omega})})})\)

The limit of BEAF is L[&]L[&]L...L[&]L, where there are n L's, and it has a limit ordinal \(\theta(\Omega^{\Omega}2)\).

Next I compare Ikosarakt1's system with FGH. X&n has growth rate \(\omega^\omega\), 2X&n=X+X&n has growth rate \(\omega^{\omega2}\), X^2&n has growth rate \(\omega^{\omega^2}\), and X^^X&n has growth rate \(\epsilon_0(=\omega\uparrow\uparrow\omega)\). In fact, any structure f(X) has growth rate \(\omega^{f(\omega)}\).And from LVO,

\(\lbrace X,X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega)\)

\(\lbrace X,2X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,1)\)

\(\lbrace X,X^2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\omega)\)

\(\lbrace X,X^3/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\omega^2)\)

\(\lbrace X,X^X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\omega^\omega)\)

\(\lbrace X,X^{X^X}/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\omega^{\omega^\omega})\)

\(\lbrace X,X\uparrow\uparrow X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\epsilon_0)\)

\(\lbrace X,\lbrace X,X/2\rbrace/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega,\theta(\Omega^\Omega))\)

\(\lbrace X,X,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+1)\)

\(\lbrace X,2X,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+1,1)\)

\(\lbrace X,\lbrace X,X,2/2 \rbrace,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+1,\theta(\Omega^\Omega+1))\)

\(\lbrace X,X,3/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+2)\)

\(\lbrace X,X,4/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+3)\)

\(\lbrace X,X,X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\omega)\)

\(\lbrace X,X,X\uparrow\uparrow X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\epsilon_0)\)

\(\lbrace X,X,\lbrace X,X,X/2\rbrace/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\theta(\Omega^\Omega+\omega))\)

\(\lbrace X,X,1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega)\)

\(\lbrace X,X,2,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega+1)\)

\(\lbrace X,X,3,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega+2)\)

\(\lbrace X,X,X,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega+\omega)\)

\(\lbrace X,X,\lbrace X,X,1,2/2 \rbrace,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega+\theta(\Omega^\Omega+\Omega))\)

\(\lbrace X,X,1,3/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega2)\)

\(\lbrace X,X,1,4/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega3)\)

\(\lbrace X,X,1,X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega\omega)\)

\(\lbrace X,X,1,\lbrace X,X,1,X/2 \rbrace/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega\theta(\Omega^\Omega+\Omega\omega)\)

\(\lbrace X,X,1,1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^2)\)

\(\lbrace X,X,2,1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^2+1)\)

\(\lbrace X,X,1,2,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^2+\Omega)\)

\(\lbrace X,X,1,1,3/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{2}2)\)

\(\lbrace X,X,1,1,1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^3)\)

\(\lbrace X,X(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^\omega)\)

\(\lbrace X,2X(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^\omega,1)\)

\(\lbrace X,X,2(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^\omega+1)\)

\(\lbrace X,X,1,2(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^\omega+\Omega)\)

\(\lbrace X,X,1,1,2(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^\omega+\Omega^2)\)

\(\lbrace X,X(1)3/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega}2)\)

\(\lbrace X,X(1)4/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega}3)\)

\(\lbrace X,X(1)X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega}\omega)\)

\(\lbrace X,X(1)\lbrace X,X(1)X/2 \rbrace/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega}\theta(\Omega^\Omega+\Omega^{\omega}\omega))\)

\(\lbrace X,X(1)1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega+1})\)

\(\lbrace X,X(1)1,1,2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega+2})\)

\(\lbrace X,X(1)(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega2})\)

\(\lbrace X,X(1)(1)(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega3})\)

\(\lbrace X,X(2)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^2})\)

\(\lbrace X,X(2)(1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^2+\omega})\)

\(\lbrace X,X(2)(2)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^{2}2})\)

\(\lbrace X,X(3)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^3})\)

\(\lbrace X,X(0,1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^\omega})\)

\(\lbrace X,X((1)1)2/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\omega^{\omega^\omega}})\)

\(\lbrace X\uparrow\uparrow X\& X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\epsilon_0})\)

\(\lbrace X\& X\& X/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\omega)})\)

\(\lbrace X,X/3 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)})\)

\(\lbrace X,X,1,2/3 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}+\Omega)\)

\(\lbrace X\& X\& X/3 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}+\Omega^{\theta(\Omega^\omega)})\)

\(\lbrace X,X/4 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}2)\)

\(\lbrace X,X/5 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}3)\)

\(\lbrace X,X/X \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}\omega)\)

\(\lbrace X,X/1,2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)+1})\)

\(\lbrace X,X/1,1,2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)+2})\)

\(\lbrace X,X/1(1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)+\omega})\)

\(\lbrace X,X/1(0,1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)+\omega^\omega})\)

\(\lbrace X,X/1/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)2})\)

\(\lbrace X,X/1/1,2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)2+1})\)

\(\lbrace X,X/1/1/2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)3})\)

\(\lbrace X,X(/1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega})\)

\(\lbrace X,X(/1)(/1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega2})\)

\(\lbrace X,X(/2)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega^2})\)

\(\lbrace X,X(/0,1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega^\omega})\)

\(\lbrace X,X(/(1)1)2 \rbrace\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega^{\omega^\omega}})\)

\(X\uparrow\uparrow X\&\& X\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\epsilon_0})\)

\(\lbrace X,X/2 \rbrace\&\& X\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)^2})\)

\(\lbrace X,2X/2 \rbrace\&\& X\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega,1)})\)

\(\lbrace X,X,2/2 \rbrace\&\& X\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega+1)})\)

\(X\&\& X\&\& X\) has growth rate \(\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)\omega})})\)

\(\lbrace X,X//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2)\)

\(\lbrace X,2X//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2,1)\)

\(\lbrace X,X,2//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+1)\)

\(\lbrace X,X,1,2//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega)\)

\(\lbrace X,X/2//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^\Omega)})\)

\(\lbrace X,X//3 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)})\)

\(\lbrace X,X//4 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)}2)\)

\(\lbrace X,X//1,2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)+1})\)

\(\lbrace X,X//1//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)2})\)

\(\lbrace X,X//2 \rbrace\&\&\& X\) has growth rate \(\theta(\Omega^{\Omega}2+\Omega^{\theta(\Omega^{\Omega}2)^2})\)

\(\lbrace X,X///2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}3)\)

\(\lbrace X,X(1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega)\)

\(\lbrace X,X/(1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}(\omega+1))\)

\(\lbrace X,X(1)//2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega2)\)

\(\lbrace X,X(1)(1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^2)\)

\(\lbrace X,X(1)(1)(1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^3)\)

\(\lbrace X,X(2)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^\omega)\)

\(\lbrace X,X(2)(1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega+1})\)

\(\lbrace X,X(2)(2)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega2})\)

\(\lbrace X,X(3)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega^2})\)

\(\lbrace X,X(4)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega^3})\)

\(\lbrace X,X(0,1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega^\omega})\)

\(\lbrace X,X((1)1)/2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega}\omega^{\omega^{\omega^\omega}})\)

\(\lbrace L,X\uparrow\uparrow X \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega}\epsilon_0)\)

\(\lbrace L,L \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega}\theta(\Omega^\Omega))\)

\(\lbrace L,X,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega+1})\)

\(\lbrace L,X,3 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega+2})\)

\(\lbrace L,X,X \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega+\omega})\)

\(\lbrace L,X,L \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega+\theta(\Omega^\Omega)})\)

\(\lbrace L,X,1,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega2})\)

\(\lbrace L,X,2,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega2+1})\)

\(\lbrace L,X,1,3 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega3})\)

\(\lbrace L,X,1,X \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega\omega})\)

\(\lbrace L,X,1,1,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^2})\)

\(\lbrace L,X,1,2,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^2+\Omega})\)

\(\lbrace L,X,1,1,3 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{2}2})\)

\(\lbrace L,X,1,1,1,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^3})\)

\(\lbrace L,X(1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\omega})\)

\(\lbrace L,2X(1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\omega}2)\)

\(\lbrace L,X,2(1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\omega+1})\)

\(\lbrace L,X,1,2(1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\omega+\Omega})\)

\(\lbrace L,X(1)3 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega}2})\)

\(\lbrace L,X(1)4 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega}3})\)

\(\lbrace L,X(1)X \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega}\omega})\)

\(\lbrace L,X(1)1,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega+1}})\)

\(\lbrace L,X(1)1,1,2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega+2}})\)

\(\lbrace L,X(1)(1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega2}})\)

\(\lbrace L,X(2)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega^2}})\)

\(\lbrace L,X(0,1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega^\omega}})\)

\(\lbrace L,X((1)1)2 \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^{\omega^{\omega^\omega}}})\)

\(X\uparrow\uparrow X@X\) has growth rate \(\theta(\Omega^{\Omega^{\epsilon_0}})\)

\(\lbrace X,X/2 \rbrace@X\) has growth rate \(\theta(\Omega^{\Omega^{\theta(\Omega^\Omega)}})\)

\(X@X@X\) has growth rate \(\theta(\Omega^{\Omega^{\theta(\Omega^{\Omega^\omega})}})\)

\(\lbrace X,X\backslash 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega})\)

\(\lbrace X,2X\backslash 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega},1)\)

\(\lbrace X,X,2\backslash 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+1)\)

\(\lbrace X,X/2\backslash 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^\Omega)})\)

\(\lbrace X,X\backslash 3 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})})\)

\(\lbrace X,X\backslash 4 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})}2)\)

\(\lbrace X,X\backslash 1,2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})+1})\)

\(\lbrace X,X\backslash 1\backslash 2\rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})2})\)

\(\lbrace X,X(\backslash 1)2\rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})\omega})\)

\(X\uparrow\uparrow X@@X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})\epsilon_0})\)

\(\lbrace X,X\backslash 2\rbrace@@X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega})^2})\)

\(\lbrace X,2X\backslash 2\rbrace@@X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega},1)})\)

\(X@@X@@X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\theta(\Omega^{\Omega^\Omega}+\Omega^{theta(\Omega^{\Omega^\Omega})\omega})})\)

\(\lbrace X,X\backslash\backslash 2\rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^\Omega)\)

\(\lbrace X,X\backslash\backslash\backslash 2\rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega}2)\)

\(\lbrace X,X(1)\backslash 2\rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega}\omega)\)

\(\lbrace L2,X\uparrow\uparrow X\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega}\epsilon_0)\)

\(\lbrace L2,L2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega}\theta(\Omega^{\Omega^\Omega}))\)

\(\lbrace L2,X,2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega+1})\)

\(\lbrace L2,X,3\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega+2})\)

\(\lbrace L2,X,1,2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega2})\)

\(\lbrace L2,X,1,3\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega3})\)

\(\lbrace L2,X,1,1,2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^2})\)

\(\lbrace L2,X(1)2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\omega})\)

\(\lbrace L2,X(0,1)2\rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^{\omega^\omega}})\)

\(X\uparrow\uparrow X%X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\epsilon_0})\)

\(X%X%X\) has growth rate \(\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\theta(\Omega^{\Omega^\Omega}+\Omega^{\Omega^\omega})})\)

\(\lbrace X,X | 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}2)\)

\(\lbrace X,X - 2 \rbrace\) has growth rate \(\theta(\Omega^{\Omega^\Omega}3)\)

\(\lbrace LX \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}\omega)\)

Use square-bracketed arrays below.

\(\lbrace [L,L] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega}\theta(\Omega^\Omega))\)

\(\lbrace [L,X,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+1})\)

\(\lbrace [L,2X,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+1}2)\)

\(\lbrace [L,X^2,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+1}\omega)\)

\(\lbrace [L,X,3] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+2})\)

\(\lbrace [L,X,X] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\omega})\)

\(\lbrace [L,X,1,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega})\)

\(\lbrace [L,X,2,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega+1})\)

\(\lbrace [L,X,1,3] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega2})\)

\(\lbrace [L,X,1,1,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^2})\)

\(\lbrace [L,X,1,1,1,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^3})\)

\(\lbrace [L,X(1)2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^\omega})\)

\(\lbrace [L,X(1)1,2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\omega+1}})\)

\(\lbrace [L,X(1)(1)2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\omega2}})\)

\(\lbrace [L,X(2)2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\omega^2}})\)

\(\lbrace [L,X(0,1)2] \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\omega^\omega}})\)

\(\lbrace X\uparrow\uparrow X[\&]L \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\epsilon_0}})\)

\(\lbrace [X,X/2][\&]L \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})\)

\(\lbrace [X,X/2][\&]L[\&]L \rbrace_{X,X}\) has growth rate \(\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^{\Omega^\Omega+\Omega^{\theta(\Omega^\Omega)}})}})\)

So Ikosarakt1's system has limit ordinal \(\theta(\Omega^{\Omega^{\Omega}2})\).

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