Can a (relatively consistent) cardinal notion be given so that its usual Mahlo operation is (probably at least) not consistent?
2026-03-25 20:06:35.1774469195
Mahlo operation, consistency border
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Sure - "is a successor ordinal." For any (fine, uncountable) infinite cardinal $\kappa$, the set of limit cardinals below $\kappa$ is a club which avoids the set of successor ordinals.
I suspect, though, that this isn't really what you want. So in the opposite direction, let me observe:
This is an awkward thing to claim since it's inherently hard (if not impossible) to justify, but it is to the best of my knowledge true.