User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-35392788-20180807165603

For that matter, we could add the metastage cardinal X, which is to T what T is to M. The ideal case would be to add the "superstage cardinal", C(1{1,0}0), but I wanted to say something else.

It is only now that I realize that UNOCF is pretty much incompatible with well-defined large cardinals. The C function made us think that we could just push out extensions ad infinitum. The thing is, there are only so many large cardinal properties. It's easy to get up to C(1;#;#) because of the concepts of a-inaccessibility and a-Mahloness. But after that, we get stuck because there is no such thing as a-weakly compactness. Even if you were to solve that, what about C(1;0;0;0)? And don't even get me started on the stage cardinal.

Is it possible to make an ordinal notation that acts like UNOCF? Surely. Is it possible to make an actual OCF? I don't think so, because an OCF is purely set-theoretical. Just look at how insane the definition up to I(1@w) is. Nish said that he was going to try making an ordinal notation that acts like UNOCF. In the meantime, you can pretty much forget about seeing UNOCF being well-defined as an actual OCF.