Hi! I’ve been making an OCF utilising degrees of Mahloness following the following fashion:
- If the cardinal which the input is being collapsed below is regular but not weakly Mahlo etc, then the output will be a limit ordinal.
- If the. cardinal which the input is being collapsed below is weakly Mahlo but not weakly 1-Mahlo etc, then the output will be a regular cardinal.
- If the cardinal which the input is being collapsed below is weakly 1-Mahlo but not weakly 2-Mahlo etc, then the output will be a weakly Mahlo cardinal,
- …
- If the cardinal which the input is being collapsed below is weakly \(\omega\)-Mahlo but not weakly \(\omega + 1\)-Mahlo etc, then the output will be a weakly pseudo-\(\omega\)-Mahlo cardinal (\(\kappa\) is weakly pseudo-\(\omega\)-Mahlo if it is weakly \(n\)-Mahlo for all \(n \in \omega\)).
But what should I do if the cardinal which the input is being collapsed below is weakly pseudo-\(\omega\)-Mahlo? It’ll certainly be valid for collapse, since it’s regular, Mahlo etc, but the “Mahloness degree” is a limit ordinal, so we can’t just return a cardinal which is directly stationary in it.