User blog comment:Alemagno12/About the fanmade OCF/@comment-32697988-20180408234329/@comment-30754445-20180409003308

I assume this means that the proposed notations should be able to write any countable ordinal up to PTO(KP+Pi3 reflection), which is the usual limit for OCF's that use K. That is what people usually mean when they say "weakly compact level".

So no, just naively replacing every Ω with K obviously won't cut it, because such a notation doesn't actually use the fact that K is weakly compact. Basically we have this hierarchy of complexity:

The K (weakly compact) functions as a "wildcard" that tracks the various types of Mahlos (M's).

The Mahlos (M's) are "wildcards" that track the various types of inaccessibles (I's).

The Inaccessibles (I's) are there to track the ever increasing constructive functions of the various Ω's.

The Ω's, of-course, are there to track ever increasing fixed points of countable ordinals, which we then use to track the levels of recrusion we're using to build actual finite numbers.

It's a huge complicated structure, and any notation that is said to "reach the weakly compact level" is expected to be able to handle it in full.