User blog comment:Ecl1psed276/BM2 Analysis - A Summary/@comment-30754445-20180709051740/@comment-25601061-20180713042229

@P bot:

First, what are you referring to with ordinal notation and ordinal function ? And second, you don't need inaccessibles to define an OCF as strong as other standard inaccessible OCFs - in fact, you can just use nonrecursive ordinals and get the same results.

@PsiCubed2:

First, how did you do the survey? I don't see anything about the survey on your blog posts, and I checked Hyp cos's talk page and nothing about the survey was there either.

And second, Hyp cos and Deedlit might be the top googologists on the wiki and yet only understand up to weakly compacts, but on Discord we've advanced a lot further.

The behaviour of Rathjen's (the standard) Mahlo and weakly compact OCF is actually very easy to understand, yet its definition makes it so much harder, as you (1) need to understand how the cardinals are defined and (2) even if you don't understand their definitions and you're just told that they're big enough for their role, you still have to figure out how the definition works and do examples and analysis until you understand what the cardinals are actually supposed to do. It's just like trying to understand pDAN or DAN merely using their definition.

Deedlit actually tries to explain this in his blog posts, but Username (with UNOCF) (1) skips over the actual definition of the cardinals, (2) gives a more straightforward definition, and (3) explains the behaviour of the cardinals in a better way, going more in-depth in the explanation and giving more examples - and, even though UNOCF's K is weaker than Rathjen's K, UNOCF still goes further than his weakly compact OCF with stage and meta^x-stage cardinals, with Rathjen's K being equal to ψT(TT T ) in UNOCF.

The people on the Discord have learned UNOCF in its entirety without much difficulty, and some people have actually tried to go higher - Ecl1psed and Aarex have made extensions to meta^x-stage cardinals that far surpass UNOCF (although these are mostly concepts and I'm probably the only one that's familiar with the specific collapse of (1,0)-ex-metastages, one of the smallest of the extended cardinals, let alone higher extended cardinals like Mu and hyperspace cardinals) and Aarex has made his own OCF called TAOCF with stronger collapses of cardinals beyond Mahlos that also greatly surpasses UNOCF.

And yes, ordinals at these levels have been studied and used in analysis - for example, I understand the behaviour of and have made analysis of BM2, and the limit of Trio Sequence System using the collapse of the extensions of meta^x-stages I mentioned above is ψ(ΓE+1) where E is the first (1,0)-ex-metastage (which might seem strange for a milestone ordinal like that, but it actually makes sense once you consider how E behaves).

And all of this was made by the people on the googology Discord, most of which probably haven't even graduated yet.

In short, '''high-level googology is not that hard. '''You don't necessarily need to have a degree in mathematics to understand how weakly compacts are collapsed - you just need a good and simple explanation. Sadly, only a few of those explanations exist for ordinals up to weakly compacts, and none exist for higher ordinals - even people like Deedlit or Hyp cos, which I believe are at college level math, don't understand OCFs that collapse ordinals higher than weakly compacts. Most of the explanations and definitions in the papers are kept formal and are intended only for the people who understand the level of math used in the OCFs (as you would expect from mathematical papers) - and, unfortunately, there are no such people in the googological community.

tl;dr we do know what we're doing, you're just out of the loop