## FANDOM

10,821 Pages

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 $$\theta(\Omega^{\Omega^{\Omega}2})$$.