User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-30754445-20180807223501

"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."

That's actually really easy to fix, because the cardinals used in UNOCF are an overkill anyway.

There's really no need for UNOCF's K to be a weakly compact, or for UNOCF's T to be a stage cardinal. Since the whole thing isn't formally defined anyway, you can just imagine that T (and X and whatever additional layers you want) are somewhat smaller than their current value, yet large enough for the system to work.

So this is not the problem. The problem is that sooner or later, the additional extensions will stop making the system any stronger. If, say, UNOCF is modelled after Deedlit's weakly compact cardinal notation, then it won't be able to go much higher than standard weakly compact OCFs. You can add another million types of cardinals and it won't change a thing. Unless you (or in this case - Username5243) are an expert on set theory, you are limited by the strength of the original notation you're basing your own version on.

"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."

Actually, the two concepts are one and the same.

If you could make an ordinal notation that acts like UNOCF and is actually well defined, then it would be an actual OCF.

Why? Because the notation itself allows us to manipulate the M's and the K's and T's and the X's as cardinals. For all intents and purposes, they are cardinals.

Now, it might be difficult to find the proper actual cardinals that would work for the various M's and K's and T's and X's, but it should be possible in principle.

"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."

The former is probably impossible without the latter.

How, on earth, do you plan to prove that this "ordinal notation" works as intended, without collapsing actual cardinals?