User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108154849/@comment-39541634-20200109111238

@Tetramur

"X^(X^^X+3) = X^(X^^X)*X^3 = X^^X*X^3 (because X^^X is infinite, and X^(X^^X) = X^^X)"

Precisely.

We are allowed to do such manipulations when X is infinite. In fact, what you've just written is nothing more than the ordinal equation:

ω^(ε₀+3) = ε₀*(ω^3).

So yes, when we are talking about ordinals, this equation is true.

But in BEAF, we're talking about a finite structure that lives in pentational space, which we're supposed to get by replacing the X's with p's.

Say p is equal to 3. Then we have:

X^^X & 3 = 3^^3 & 3 = a superdimensional array of size 3, containing 7625597484987 threes.

X^(X^^X) & 3 = 3^(3^^3) & 3 = 3^^4 & 3 = a tridimensional array of size 3, containing 3^7625597484987 threes.

See what happens? The two expressions do not correspond to the same structure. Not only do they not have the same number of entries, but they end up expanding into arrays of two completely different conceputal levels: The first one is merely superdimensional, while the second one is tridimensional.

(note that both of these results are well-defined numbers)

@P-bot

"Then my metaphor on the difference of n→f_{ω↑↑ω}(n) and n→f_{n↑↑n}(n) might not be far from the situation"

Kinda.

The actual situation is more subtle than that.

How much do you know about BEAF? 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?

"At least, I guess that the OP states that Bowers' intended one satisfies the single-step conversion X^^^X→p^^^p which looks intuitively very weak, while the OP states that Bowers' intended one belongs to BEAF-BSC. It should imply that Bowers' method fits the table. But... is it so strong that X^^^X corresponds to Γ_0?"

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

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