User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-30004975-20180116061613/@comment-30754445-20180409013701

@CatIsFluffy, Regarding the diagonalization over I,M and K:

I'm not sure what indescribables do exactly, but you can diagonalize over I,M,K... This would get us to something called "stable ordinals", which - as far as I know - is the absolute highest level that the pros have a well-studied notation for.

The catch is that doing such a "diagonalization" is... well... difficult. The main challenge here is that every level is fundamentally more complicated than the previous one. You can't just diagonalize over I,M,K,... with a single kind of formula, because the whole point of "inaccessibles" and "mahlos" and "weakly compact" is that each one represents a much higher level of abstraction.

Case in point: Look how radically different the last 3 sections (IV, V, VI) of Deedlit's blog series are from one another. There's no obvious trend which we can use here to "extrapolate" even one level higher (to Pi4-reflections), let alone a clear pattern we can "diagonalize" to infinity.

Some pattern, of-course, does exists. This diagonalization has been done before (by Stegert, I believe). But it is quite a tough cookie to crack, and it is orders of magnitude more complicated than anything we've ever seen on this wiki (which is probably why Deedlit hasn't yet published part VII).