User blog comment:Deedlit11/Ordinal Notations III: Collapsing Higher Cardinalities/@comment-7484840-20130706154845/@comment-5529393-20130709041427

$$/vartheta(\Omega_{\Omega_3})$$ is not the same as $$/vartheta(\Omega_{\Gamma_{\Omega_2 + 1}})$$. Actually, based on my definition above, $$/vartheta(\Omega_3)$$ is larger than $$/vartheta(\Gamma_{\Omega_2 + 1})$$. This is because $$/vartheta_2(1) = \Gamma_{\Omega_2 + 1})$$, whereas $$/vartheta(/Omega_3) = \vartheta(\alpha)$$ where $$\alpha$$ is the smallest ordinal such that $$\alpha = \vartheta_2 (\alpha)$$.

Moreover, $$/vartheta(\Omega_{\Omega_3})$$ is larger than $$/vartheta(\Omega_{/vartheta(\Omega_3)})$$, as with $$C(\Omega_3, 0)$$ we are only allowed to use  $$\vartheta_\alpha(\beta)$$ for $$\alpha = 0, 1$$, but with $$C(\Omega_{\Omega_3}, 0)$$ we are allowed to use $$\vartheta_\alpha(\beta)$$ for any $$\alpha < \Omega_3 $$.