## FANDOM

10,821 Pages

In Bird's array notation, he reach a $$\vartheta(\epsilon_{\Omega+1})$$, for now $$\vartheta(\vartheta_1(\Omega))$$ and he beat me, so extension of hyper-nested arrays.

Note : $$¬$$ as short one, $$\backslash_2$$.

$$[1 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})$$

$$[2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+1$$

$$[1,2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\omega$$

$$[1 [2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\omega^\omega$$

$$[1 [1 \backslash 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\varepsilon_0$$

$$[1 [1 \backslash 1 \backslash 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\zeta_0$$

$$[1 [1 \backslash 1 \backslash 1 \backslash 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\eta_0$$

$$[1 [1 [2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\omega)$$

$$[1 [1 [2 ¬ 2] 1 [2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\omega 2)$$

$$[1 [1 [3 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\omega^2)$$

$$[1 [1 [1,2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\omega^\omega)$$

$$[1 [1 [1 [2] 2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\omega^{\omega^\omega})$$

$$[1 [1 [1 \backslash 2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\varepsilon_0)$$

$$[1 [1 [1 [1 \backslash 2 ¬ 2] 2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\vartheta(\varepsilon_0))$$

$$[1 [1 [1 [1 [1 \backslash 2 ¬ 2] 2 ¬ 2] 2 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\vartheta(\vartheta(\varepsilon_0)))$$

$$[1 [1 [1 ¬ 3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega)$$

$$[1 [1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega 2)$$

$$[1 [1 [2 ¬ 3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega\omega)$$

$$[1 [1 [1 \backslash 2 ¬ 3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega\vartheta(1))$$

$$[1 [1 [1 [1 ¬ 2] 2 ¬ 3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega\vartheta(\Omega))$$

$$[1 [1 [1 ¬ 4] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^2)$$

$$[1 [1 [1 ¬ 5] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^3)$$

$$[1 [1 [1 ¬ 1,2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^\omega)$$

$$[1 [1 [1 ¬ 2,2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^{\omega +1})$$

$$[1 [1 [1 ¬ 1,3] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^{\omega 2})$$

$$[1 [1 [1 ¬ 1 \backslash 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^{\varepsilon_0})$$

$$[1 [1 [1 ¬ 1 ¬ 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^\Omega)$$

$$[1 [1 [1 [1 \backslash_3 3] 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(\Omega^{\Omega^\Omega})$$

$$[1 [1 [1 [1 \backslash_3 1 \backslash_3 2] 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(^4\Omega)$$

$$[1 [1 [1 [1 [1 \backslash_4 3] 2] 2] 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})+\vartheta(^5\Omega)$$

$$[1 [1 \backslash_{1,2} 2] 2 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1})2$$

$$[1 \backslash_{1,2} 3]$$ has level $$\vartheta(\epsilon_{\Omega+1},1)$$

$$[1 \backslash_{1,2} 4]$$ has level $$\vartheta(\epsilon_{\Omega+1},2)$$

$$[1 \backslash_{1,2} 1 \backslash 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+1)$$

$$[1 \backslash_{1,2} 1 [2 ¬ 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+\omega)$$

$$[1 \backslash_{1,2} 1 [1 ¬ 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+\Omega)$$

$$[1 \backslash_{1,2} 1 [1 ¬ 1 ¬ 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+\Omega^\Omega)$$

$$[1 \backslash_{1,2} 1 [1 [1 \backslash_3 3] 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+^3\Omega)$$

$$[1 \backslash_{1,2} 1 [1 [1 \backslash_3 1 \backslash_3 2] 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}+^4\Omega)$$

$$[1 \backslash_{1,2} 1 \backslash_{1,2} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}2)$$

$$[1 [2 \backslash_{2,2} 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}\omega)$$

$$[1 [1 \backslash_{2,2} 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}\Omega)$$

$$[1 [1 \backslash_{2,2} 1 \backslash_{2,2} 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}\Omega^\Omega)$$

$$[1 [1 [1 \backslash_{2,2} 3] 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*^3\Omega)$$

$$[1 [1 [1 \backslash_{2,2} 1 \backslash_{2,2} 2] 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*^4\Omega)$$

Now I am $$\backslash_{1,3}$$.

$$[1 \backslash_{1,3} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1})$$

$$[1 \backslash_{1,3} 1 \backslash_{1,3} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}2)$$

$$[1 [1 \backslash_{2,3} 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}\Omega)$$

$$[1 [1 \backslash_{2,3} 1 \backslash_{2,3} 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}\Omega^\Omega)$$

$$[1 [1 [1 \backslash_{3,3} 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*^3\Omega)$$

Continue that:

$$[1 \backslash_{1,n+1} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*...*\epsilon_{\Omega+1})$$ (n $$\epsilon_{\Omega+1}$$'s)

$$[1 \backslash_{1,n+1} 1 \backslash_{1,3} 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*...*\epsilon_{\Omega+1}2)$$ (n $$\epsilon_{\Omega+1}$$'s)

$$[1 [1 \backslash_{2,n+1} 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*...*\epsilon_{\Omega+1}\Omega)$$ (n $$\epsilon_{\Omega+1}$$'s)

$$[1 [1 \backslash_{2,n+1} 1 \backslash_{2,n+1} 2] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*...*\epsilon_{\Omega+1}\Omega^\Omega)$$ (n $$\epsilon_{\Omega+1}$$'s)

$$[1 [1 [1 \backslash_{3,n+1} 3] 2]$$ has level $$\vartheta(\epsilon_{\Omega+1}*\epsilon_{\Omega+1}*...*\epsilon_{\Omega+1}*^3\Omega)$$ (n $$\epsilon_{\Omega+1}$$'s)

Ok, let $$[1 \backslash_A 2]$$, when A:

$$1,1,2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega})$$

$$1,2,2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega +1})$$

$$1,1,3$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega 2})$$

$$1,1,1,2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega^2})$$

$$1,1,1,1,2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega^3})$$

$$1 [2] 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega^\omega})$$

$$1 [1,2] 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\omega^{\omega^\omega}})$$

$$1 \backslash 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\epsilon_0})$$

$$1 \backslash 1 \backslash 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\zeta_0})$$

$$1 \backslash 1 \backslash 1 \backslash 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\eta_0})$$

$$1 [2 ¬ 2] 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\vartheta(\omega)})$$

$$1 [1 ¬ 3] 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\vartheta(\Omega)})$$

$$1 \backslash_{1,2} 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\vartheta(\epsilon_{\Omega+1})})$$

$$1 \backslash_{1 \backslash_{1,2} 2} 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^{\vartheta({\epsilon_{\Omega+1}}^{\vartheta(\epsilon_{\Omega+1})})})$$

Also $$[1 \backslash_\alpha 2]$$, $$\alpha \rightarrow 1 \backslash_\alpha 2$$ has level $$\vartheta({\epsilon_{\Omega+1}}^\Omega)$$

Chris Bird will make a rules of notation, but he reach over $$\vartheta(\epsilon_{\Omega+1})$$!