User blog comment:Hyp cos/Question about weak compactness/@comment-35470197-20180911215601/@comment-35392788-20180914171546

I mean, that would obviously be well-defined, but I think we are going for strength and consistency at the same time. Why do I believe such an extension is inconsistent ?

Well, for example, (weakly) Mahlos can express any level of (recursive?) inaccessibility over cardinals below Mahlos, as in, they can't talk about regular limits of a set of Mahlos, although they can talk about regular limits of a set of inaccessibles greater than some Mahlos, for example. To do that, we need 2-Mahlos.

Likewise, weakly compacts can express any level of (recursive?) Mahloness over cardinals below weakly compacts. Hence, a "2-weakly compact" should be able to talk about things like :

- "the smallest weakly compact that is a limit of weakly compacts" (your definition of a 2-compact, and what i'd call an "inaccessible weakly compact")

- "the smallest cardinal such that the set of weakly compacts below it is stationary in itself" (what i'd call a "weakly compact Mahlo")

- "the smallest cardinal such that the set of 'compact Mahlos' below it is stationary in itself" (what i'd call a "weakly compact 2-Mahlo")

- "the smallest weakly compact such that the set of weakly compacts below it is stationary in itself" (what i'd call a "Mahlo weakly compact")