User blog comment:Ubersketch/Large Cardinals vs. Non-recursive Ordinals vs. Stages/@comment-31580368-20190108065600

The problem with using large cardinals is that, as Taranovsky wrote, "сardinals structure beyond n-Subtle cardinal is rich but poorly understood". Taranovsky attempted to describe it in terms of Reflective Cardinals. Before that, Rathjen created OCF based on extendible cardinals, which he redefined as reducible cardinals. This OCF could express countable ordinals above the level Δ13-CA. The definition of this notation takes about 30 pages. But J.C.Stegert showed that this definition contains some technical errors.

Stage cardinals don't have the correct mathematical definition therefore OCF with Stage cardinals is meaningless.

Non-recursive ordinals structure known much better. But their collapsing is very complex. I tried to create a system that creates a brief definition of such ordinals - SLCON. And it turns out that even the definitions of such ordinals in short form themselves are similar to a collapsing function.

Well, the main difficulty, the more largest things we collapse, the more difficult it becomes to create ordinal notation based on this OCF. As P進大好きbot wrote, OCF and ordinal notation based on it is not the same. For example, the definition of a Stegert's OCF based on Subtle cardinal is not very complicated. Hypcos redefined this function in terms of α=(+α)-stable that can conditionally compare with n-Subtle сardinal. But ordinal notation system based on this OCF defined by Stegert, so complex that the expression on it of countable ordinals never used on this site. Anyway, very few people understand this ordinal notation system.