User blog comment:Hyp cos/Question about weak compactness/@comment-35470197-20180911215601/@comment-35392788-20180913131903

Your definition of a 2-weakly compact cardinal isn't suited for OCF purposes then. A weakly compact that is a limit of weakly compacts would be what i'd call an "inaccessibly compact" (informal) cardinal. And in my opinion, such a cardinal would be to \(K_x\) what I is to \(\Omega_x\). And I think we'd want a 2-weakly compact cardinal to be able to express things like a "Mahlo compact", i.e some k such that the set of weakly compacts below k is stationary in itself.