User blog comment:Googleaarex/Bashicu Matrix System Analysis (Part 1)/@comment-27516045-20170725132813/@comment-5529393-20170727041050

@Kyodaisuu:  do you understand BashicuHyudora's definition of his new OCF? It seems strange that he would create a new OCF and not explain it anywhere.

@HypCos: Would you be willing to further explain how pDAN compares with the I,M, and K OCF's up to the point where they reach their end? I get that {1{1,,2,,,2}2} corresponds with M and {1{1,,1,,2,,,2}2} corresponds with K, but I would definitely like to see further explanation and comparisons to see exactly how it works.

@PsiCubed2: It would be awesome if those familiar with the stronger ordinal notations (Pi-4 reflection and beyond) would give an explanation for how those work, but I guess that is unlikely at this time. I suppose we could ask.

SAN seems to go very high - it appears that we go well beyond the weakly compact OCF even with just pDAN, and Hyp Cos has seven sections beyond that. Of course, his definitions are quite lengthy and complex. This may be because he is essentially defining the fundamental sequences for the "ordinals" he is extending to, and typically defining fundamental sequences get a lot more complex than just defining the original ordinal notations. So is there a "fundamental sequence-less" version of SAN, that could be much simpler both in length and in comprehensibility? I don't know, but it could be fun to think about. Of course, Hyp Cos's main objective is to define large finite numbers, so removing the fundamental sequences is counter to what he wants to do.

Taranovsky's notation is very elegant, and seemingly very powerful. The trouble I've been having with it is, since it is defined in terms of a (very simple) comparison relation rather than via fundamental sequences, it is often hard to determine (at least for me) which major ordinal notation comes next. I guess the traditional ordinal notations aren't defined via fundamental sequences either, but I haven't had as much trouble with them. I guess it is because it is easy to tell what the permissible notations are for traditional OCFs, whereas the standard notations for TON, while fairly simply defined, are rather tricky to evaluate in practice. Hyp Cos seems to have a much better handle on this however, so perhaps it simply comes down to a matter of experience.