User blog comment:Deedlit11/Ordinal Notations III: Collapsing Higher Cardinalities/@comment-27513631-20160425221405

Could you take a look at your fundamental sequences again? Take a look at \(\vartheta(\Omega^2+\omega+1) = \varphi(2,0,\omega+1)\): You claim that \(\vartheta(\Omega^2+\omega+1)[0] = \vartheta(\Omega^2+\omega)+1 = \varphi(2,0,\omega)+1\) and \(\varphi(2,0,\omega+1)[n+1] = \vartheta(\Omega^2+\omega+1)[n+1] = \varphi(\vartheta(\Omega^2+\omega+1)[n],0) = \varphi(\varphi(2,0,\omega+1)[n])\), however the limit of this sequence is clearly \(\Gamma_{\varphi(2,0,\omega)+1}\), not \(\varphi(2,0,\omega+1)\).

(\(\vartheta\) is short for \(\vartheta_0\) here. The \(+\omega\) was used to avoid needing to pay attention to the discontinuities.)

It would be greatly appreciated if you could fix this. I believe that something like \(\alpha[n+1] = \vartheta_\nu(\beta^-+\alpha[n])\) would do the job, where \(\beta^-\) is the greatest ordinal less than \(\beta\) that is a multiple of \(\Omega_\nu\).