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## 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.