User blog comment:Deedlit11/Ordinal Notations III: Collapsing Higher Cardinalities/@comment-28606698-20170408192340

I can't understand how your last rule for FS works. I mean case when cofinality of argument is larger than cofinality of function. It seems that for any $$\beta \vartheta_\nu(\beta)[n]=\vartheta_\nu(\vartheta_\mu^n(\Omega_\mu))$$.

I would proposed $$\vartheta_\nu(\beta)[n]=\vartheta_\nu(\gamma[n])$$, where

$$\gamma[0]=\vartheta_\mu(\beta[0])$$

$$\gamma[n+1]=\vartheta_\mu(\beta[\gamma[n]])$$