User blog comment:Emlightened/Early Birthday Present For Deedlit/@comment-5529393-20170730212008/@comment-5529393-20170802040524

If you are defining $$M_\omega$$ to be the smallest $$\omega$$-weakly Mahlo cardinal, then it is weakly Mahlo by definition, and therefore regular. If you are defining $$M_\omega$$ to be the limit of some $$\omega$$ sequence, like the limit of the smallest n-weakly Mahlo cardinals for finite n, then you are right, $$M_\omega$$ will not be regular. But this is unrelated to anything I was talking about.

Similarly, if we define $$M(\alpha)$$ to enumerate the weakly Mahlo cardinals, then by definition $$M(\omega)$$ will be weakly Mahlo. But, if we define $$M(\alpha)$$ to enumerate the weakly Mahlo cardinals and limits of weakly Mahlo cardinals, then $$M_\omega$$ will not be regular. So it all depends on the definitions.