FANDOM


Bracket Notation New
















FGH

\([[0]_2]\) was \([0]_2\)  and has level \(\varepsilon_0\)

\([[0]_21]\) was \([1]_2\) and has level \(\varepsilon_1\)

\([[0]_2[0]]\) was \([[0]]_2\) and has level \(\varepsilon_{\omega}\)

\([[0]_2[[0]_2]]\) was \([[0]_2]_2\) and has level \(\varepsilon_{\varepsilon_0}\)

\([[0]_2[[0]_2[[0]_2]]]\) was \([[[0]_2]_2]_2\) and has level \(\varepsilon_{\varepsilon_{\varepsilon_0}}\)

\([[0]_2[0]_2]\) was \([0]_3\) and has level \(\zeta_0\)

\([[0]_2[0]_2[0]_2]\) was \([0]_4\) and has level \(\eta_0\)

\([[1]_2]\) was \([0]_{[0]}\) and has level \(\vartheta(\omega)\)

\([[2]_2]\) was \([0]_{[1]}\) and has level \(\vartheta(\omega^2)\)

\([[[[0]_2]]_2]\) was \([0]_{[0]_2}\) and has level \(\vartheta(\varepsilon_0)\)

\([[[[0]_2[0]_2]]_2]\) was \([0]_{[0]_3}\) and has level \(\vartheta(\zeta_0)\)

\([[[[1]_2]]_2]\) was \([0]_{[0]_{[0]}}\) and has level \(\vartheta(\vartheta(\omega))\)

\([[[[[[1]_2]]_2]]_2]\) was \([0]_{[0]_{[0]_{[0]}}}\) and has level \(\vartheta(\vartheta(\vartheta(\omega)))\)

\([[[0]_2]_2]\) has level \(\vartheta(\Omega)\) since it nests using [] bracket.

\([[[0]_2[0]_2]_2]\) has level \(\vartheta(\Omega2)\) since it nests like \(\Omega\) again.

\([[[1]_2]_2]\) has level \(\vartheta(\Omega\omega)\)

\([[[[0]_2]_2]_2]\) has level \(\vartheta(\Omega^2)\)

\([[0]_3]\) has level \(\vartheta(\Omega^\omega)\), the SVO

\([[0]_3[0]_2]\) has level \(\vartheta(\Omega^\omega+1)\)

\([[0]_3[[0]_2]_2]\) has level \(\vartheta(\Omega^\omega+\Omega)\)

\([[0]_3[0]_3]\) has level \(\vartheta(\Omega^\omega2)\)

\([[1]_3]\) has level \(\vartheta(\Omega^\omega\omega)\)

\([[2]_3]\) has level \(\vartheta(\Omega^\omega\omega^2)\)

\([[[0]_2]_3]\) has level \(\vartheta(\Omega^{\omega+1})\)

\([[[0]_2[0]_2]_3]\) has level \(\vartheta(\Omega^{\omega+2})\)

\([[[1]_2]_3]\) has level \(\vartheta(\Omega^{\omega2})\)

\([[[2]_2]_3]\) has level \(\vartheta(\Omega^{\omega^2})\)

\([[[[0]_2]_2]_3]\) has level \(\vartheta(\Omega^{\Omega})\) since it nests the \([0]_3\) bracket

\([[[[0]_2[0]_2]_2]_3]\) has level \(\vartheta(\Omega^{\Omega2})\) since it nests the \([0]_3\) bracket again

\([[[[[0]_2]_2]_2]_3]\) has leve l \(\vartheta(\Omega^{\Omega^2})\)

\([[[0]_3]_3]\) has leve l \(\vartheta(\Omega^{\Omega^\omega})\)

\([[[[[0]_2]_2]_3]_3]\) has level \(\vartheta(\Omega^{\Omega^{\Omega}})\)

\([[[[0]_3]_3]_3]\) has level \(\vartheta(\Omega^{\Omega^{\Omega^\omega}})\)

\([[0]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1})\)

\([[0]_4[0]_2]\) has level \(\vartheta(\varepsilon_{\Omega+1}+1)\)

\([[0]_4[0]_3]\) has level \(\vartheta(\varepsilon_{\Omega+1}+\Omega^\omega)\)

\([[0]_4[0]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}2)\)

\([[1]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}\omega)\)

\([[[0]_2]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^2)\) nests like Omega again. I like this!

\([[[[0]_2]_2]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^\Omega)\)

\([[[0]_3]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^{\Omega^\omega})\)

\([[[[0]_3]_3]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^{\Omega^{\Omega^\omega}})\)

\([[[0]_4]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}})\)

\([[[[0]_4]_4]_4]\) has level \(\vartheta(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}})\)

\([[0]_5]\) has level \(\vartheta(\varepsilon_{\Omega+2})\)

\([[0]_6]\) has level \(\vartheta(\varepsilon_{\Omega+3})\)

\([[0]_{[0]}]\) has level \(\vartheta(\varepsilon_{\Omega+\omega})\)

\([[0]_{[0]_21}]\) has level \(\vartheta(\varepsilon_{\Omega2})\) since it uses the [] bracket to nest.

\([[0]_{[0]_22}]\) has level \(\vartheta(\varepsilon_{\Omega3})\)

\([[0]_{[0]_2[0]_2}]\) has level \(\vartheta(\varepsilon_{\Omega^2})\)

\([[0]_{[[0]_2]_2}]\) has level \(\vartheta(\varepsilon_{\Omega^\Omega})\)

\([[0]_{[0]_3}]\) has level \(\vartheta(\varepsilon_{\Omega^{\Omega^\omega}})\)

\([[0]_{[0]_4}]\) has level \(\vartheta(\varepsilon_{\varepsilon_{\Omega+1}})\)

\([[0]_{[0]_{[0]_4}}]\) has level \(\vartheta(\varepsilon_{\varepsilon_{\varepsilon_{\Omega+1}}})\)

\([0,1]\) has level \(\vartheta(\zeta_{\Omega+1})\)

Conclusion

This is not useful for my notation. We get \([[0]_2,1] = [[[[...[[0,1],1]...],1],1],1]\) but \([[0,1],1] = [[0]_{[0]_{[0]_{...}}},1]\) so I have to remake everything but it is very likely that Extended Array Notation still don't reach a larger ordinal.

But I'm happy that I reach \(\vartheta(\zeta_{\Omega+1})\) with only 6 rules.

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