User blog comment:DontDrinkH20/Some Explanations of Certain Large Cardinals/@comment-11227630-20181019090840/@comment-35470197-20181020233250

@DontDrinkH20

Hyp cos actually knows the equivalences between \(\Pi_0^1\)-indescribability and inaccesibility, and between \(\Pi_1^1\)-indescribability and weak compactness, because such topics appear many times in his blogs. I guess that you incorrectly read what he wrote. For example, he did not say that weakly Mahlo property is equivalent to \(\Pi_0^1\)-indescribability.

For the richness, I think that he is referring to properties which ensure the strength of OCFs. The largeness alone does not ensure it. For example, there is no known OCF with rank-in-rank cardinals.

If you have an idea to construct a stronger OCF with large cardinals which you recommend, it is great for you to write the precise definition of the OCF and the proof of its strength (e.g. comparison to PTO of arithmetics and set theories). I am also interested in an actual OCF, because my strongest notation is based on proof theoretic approach (immitating Friedman's technique) but not an OCF.

We always hope to obtain a new, easy to handle, and stronger OCF. I think that you have ability to make such one :)