User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-30754445-20180808215408

@Syst3ms

You are right that actually doing this is difficult. That's what I've been saying all along.

And it is certainly possible to create ordinal notations without referencing any set-theoretic stuff. Think of BMS, for example. Even BM1 is a valid ordinal notation up to a certain point (probably a bit after (0,0)(1,1)(2,1)(3,1)).

So you don't need set theory to create a notation.You just need it to show that the notation works as intended.

"Now, I'm not one to make statements about strength of systems, but according to several people, if UNOCF were well-defined up to T, it would be as strong as some standard weakly compact OCFs."

This may true, but only if Username based his notation on an already existing notation that reaches weakly compacts. There would also be absolutely no margin for error, because even a tiny oversight will make UNOCF far weaker than its parent system.

And there's absolutely no way it would be much stronger. You can't reach the weakly-compact level just by extrapolating what you did with Mahlos. You can't reach Pi-4 reflection (the next level after weakly-compacts) just by extrapolating what you did with weakly-compacts.

Think of it in this way:

Every level has a "secret ingredient" that you need to discover before you unlock it. Without it, no amount of recursion and functions and new cardinals would get you anywhere.

And there are two ways to get these secret ingredients:

(1) Learn them from someone else who already has the knowledge.

(2) Discover them yourself, by using your own understanding and logic.

Since Username openly stated that his understanding of the relevant set theoretic topics is quite limited, it is clear that he won't be uncovering these secrets on his own any time soon.

And niether would I, by the way. Even though I consider myself to be pretty knowledgable and experienced in both googology and general mathematics, I'm nowhere near the point of being able to discover such things on my own. This is really really advanced stuff.

Any way, this is why I'm saying, that UNOCF - in the best of best case scenarios - cannot be significantly stronger than the notations it was based on.

(and I highly doubt UNOCF is based on anything stronger than standard K, because there aren't any nonexpert-friendly references for anything beyond that. For Pi-4 reflections and beyond, the opportunites for following option #1 are very limited as well)