User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108154849/@comment-35470197-20200109124507

> How much do you know about BEAF?

What I think that I have ever understood is BEAF up to ε_0, but it might be a wrong understanding, because it is the first time I understood ε_0. (It might sound weird, but I started studying countable ordinals right after I started studying googology.)

> Do you understand where the power of tetrational arrays come from, from an intuitive-geometric perspective? How the structures manage to perfectly mimic Cantor Normal Forms?

I have never understood such a correspondence. Honestly, I could not understand the article on BEAF at all. When I studied BEAF up to ε_0, I followed the explanation in the introduction article of BEAF instead. I tried to creat a 1-to-1 correspondence between ordinals below ω^β and maps β→ω whose values are zero for all but finitely many inputs. After that, I tried to understand that ε_0 is the first fixed point of exponentiation, and hence we can regard ordinals below ε_0 are maps from ε_0 itself to ω whose values are zero for all but finitely many inputs. Through the interpretation, I constructed a 1-to-1 correspondence between BEAF up to ε_0 (regarded as nested arrays due to the introduction page) and ordinals below ε_0 (regarded as "nested maps" through the bijection above). So this is not based on the understanding of the notion of "structures", but is just a correspondence between nested arrays and "nested maps".

> Personally, I would guess that X↑↑↑X should correspond to ζ₀.

> If this seems surprisingly strong to you, remember that p↑↑↑p isn't just a number here.

If it is just ζ_0, then it is not so surprising, because there are many (non-unique) ways to define transfinite hyperoperators in that way. (I do not know whether mutiplearrows in structures in BEAF actually work as transfinite hyperoperators, though.)

> It represents both the number of entries and the structure they inhabit. In this case, p↑↑↑p is a "pentational cube" with p dimensions and side p.

I understood that p↑↑↑p is not the resulting value, but I am not certain about how it works in that strobg way. If the resulting structure does not include repetition of X↑↑, I have no idea on how it corresponds to ζ_0. (But this is obviously due to the lack of my understanding of the precise intension of the notion of "structures" of BEAF. Therefore this is what I should understand by myself before asking too much. At least, arguments between you and the OP are much helpful for me to guess the precise behaviour than the articles.)