FANDOM


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 \(\vartheta(\epsilon_{\Omega+1})\)!

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