User blog comment:Hyp cos/Question about weak compactness/@comment-27513631-20180916103927

I'm not sure if there's a satisfactory definition of "Weakly Compact over a set of ordinals" in such a way that the WCCs are precisely the cardinals which are "Weakly Compact over the set of smaller regular cardinals".

For weak inaccessibility and weak Mahloness, the cardinals give rise to operators on sets - in particular, the set of weak inaccessibles/Mahlos \(<\alpha\) are the ordinals in the set \(O(\text{Reg}\cap\alpha)\), where \(O\) is either the set of limits in the original set (weak inaccessible) or the Mahlo operation (weakly Mahlo).

In particular, the operation doesn't care about cardinal arithmetic - the weakly inaccessibles and weakly Mahlos don't need to be strong limit ordinals. (Even Greatly Mahlos don't need to be strong limits, and they can replace WCCs in the $\Pi_3$=reflection OCF.) However, WCCs are always strong limits, so it seems doubtful to me that there's a similar characterisation.