User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-30754445-20180806210255

"If you have an alternative definition I'll take it, but the C function is meant to go much, much further than just Mahlos. Keep in mind that I don't care in the slightest about power nor consistency with standard OCFs, I just want a formal version of UNOCF."

There are quite a few problems with what you just said.

The biggest one is this:

If UNOCF Mahlos aren't as powerful as standard OCF Mahlos, then the entire system breaks down. It is inconsistent with itself, because there are plenty of places where Username5243 implicitly assumes otherwise. This is absolutely clear from everything he wrote.

Indeed, he even explicitly claims (in the "intro page" you've linked to) that his Mahlos and weakly-compacts are precisely as powerful as the usual ones.

If they aren't, then we have a system that makes absolutely no sense. Not because it contradicts some "standard" but because it contradicts itself.

"And yes, I am very conscious that UNOCF's Mahlos are weaker than standard OCF's (same for weakly compacts, especially for those) but it's not like it's the first time Username breaks tradition."

Username did not "break any tradition" here.

It was his intention that UNOCF's M and K will be precisely as powerful as the standard version (with a possible offset of one additional "M/K" in the power-tower, which has absolutely no relevance to our current discussion).

This is obvious from both his comparisions to SAN and his comparisions to "standard OCF" in his intro page. And actually, I'm not sure what he tried to do with the C function, given that he is quite aware of the fact that his Mahlo-level collpasing function is stronger than his Mahlo-level C. He correctly stated that C(1,0,0,0,...) corresponds to Psi(M^M^w) and then (correctly again) mentioned that we can go higher with things like Psi(M^M^M).

So either the C-function is just a convenient shorthand that's not really needed for the definitions, or Username5243 was just confused and he messed his system up.

Now, you can certainly create an OCF where M and K are weaker, but it would no longer be UNOCF (formal or otherwise).

"I know that large cardinals above Mahlos get pretty weak in UNOCF. But accordingto Nish, the fact that it can be extended much MUCH further than most OCFs using the C function compensates for that."

Actually, they can't.

Even if we ignore, for a moment, the fact that this weakness results in a basic inconsistency of UNOCF itself, the fact remains that the C function alone is very very weak. It's just a simple array notation. A Veblen variant which was extended a bit. I doubt very much that a system like this could reach standard Mahlos, and I'm 100% sure it won't get anywhere near standard compact cardinals.

There's something important you should keep in mind here:

Every letter in the standard OCF (I,M,K,...) corresponds to a completely different level of doing things. Rathjen's K does a crazy thing called "Pi3-reflections" which is so beyond everything that Username5243's did with the C function, that it isn't even funny.

Kinda reminds me of all those newbies who post a salad of simple recursions and think that they've created "the largest number evar!!!!1111", or that guy who asked if his notation is as strong as HAN... You don't need to actually analyze such attempts, to know that they result in functions far weaker than even ε0.