User blog comment:Edwin Shade/How Do I Evaluate BEAF Arrays In Two Dimensions ?/@comment-30754445-20170827183955/@comment-5529393-20170830051042

I believe the basic reason why tetrational arrays work out nicely is because the ordinals (which is more or less what these array structures that keep reducing until they terminate after a finite number of steps) below $$\varepsilon_0$$ are uniquely represented in "Iterated Cantor Normal Form":  an ordinal $$\alpha > 0$$ can be uniquely expressed as a sum $$\omega^{\alpha_1} + \omega^{\alpha_2} + \cdots + \omega^{\alpha_n}$$ with $$\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n$$, and then those ordinals $$\alpha_i$$ can themselves be written as sums of powers of $$\omega$$, until everything reaches 0. So exponentiation turns out to be a very nice operation to express ordinals with. However, tetration and higher Knuth arrow operations don't seem to work out so well. For starters, if we define tetration in the obvious way:

$$\alpha \uparrow\uparrow 1 = \alpha$$

$$\alpha \uparrow\uparrow (\beta + 1) = \alpha \uparrow (\alpha \uparrow\uparrow \beta)$$

For limit $$\beta, \alpha \uparrow\uparrow \beta = \sup_{\gamma < \beta} \alpha \uparrow\uparrow \gamma$$

then

$$\omega \uparrow\uparrow \omega = \varepsilon_0$$

$$\omega \uparrow\uparrow (\omega+1) = \omega^{\varepsilon_0} = \varepsilon_0$$

$$\omega \uparrow\uparrow (\omega+2) = \omega^{\varepsilon_0} = \varepsilon_0$$

and so on; so $$\omega\uparrow\uparrow \alpha$$ will be $$\varepsilon_0$$ for any $$\alpha \ge \omega$$. Obviously this is no good. A solution that I like is to use Down-arrow notation, which does keep going up and up, however this doesn't match up with Arrow notation for finite ordinals, and so it clearly wasn't what Bowers had in mind. Another suggestion was to use something like

$$\alpha \uparrow\uparrow (\beta + 1) = (\alpha \uparrow (\alpha \uparrow \uparrow \beta)) \uparrow \alpha $$

which would also work, but again would not match up with up-arrow notation for the finite ordinals. Another option is to use the original rule, except when $$\beta$$ is of the form $$\omega \gamma + 1$$, where we do something to make sure it increases. This works, although it seems rather unnatural. Sbiis Saibian has his own notion for Knuth arrows on ordinals that seems to me very complicated. The problem with these solutions is that we don't get a nice normal form theorem like we do with addition and exponentiation; for a nice normal form theorem, we should use the Veblen function, but that doesn't seem to match up with Knuth arrows at all.

We can still define a notation for pentational arrays; for example, see my solution at http://googology.wikia.com/wiki/User_blog:Deedlit11/A_rigorous_definition_for_pentational_arrays. I think it works reasonably enough, but it certainly is not perfect; for example, $$X \uparrow\uparrow (X+1)$$ reduces to $$(X \uparrow\uparrow X)^n$$, which Sbiis objected to because $$(n \uparrow\uparrow n)^n$$ does not equal $$n \uparrow\uparrow (n+1)$$. This is true, but the problem is that there is no expression in my notation that is greater than the $$(X \uparrow\uparrow X)^n$$ and less than $$X \uparrow\uparrow (X+1)$$. (We could get more expressions by allowing arbitrary expressions to be the base of ^ or $$\uparrow\uparrow$$, but that opens up all new complications, and still wouldn't get us expressions that look like they approach $$X \uparrow\uparrow (X+1)$$.  Basically, what we want is $$X \uparrow X \uparrow \cdots X \uparrow n$$ with X X's, but there's no way to express that except in that clumsy way with ellipses.)  Sbiis has his own ideas for what Bowers' notation should be, but I don't think he has ever actually defined pentational arrays. A major stumbling block is that he feels that an important, major feature of Bowers' notation is that the prime block of the X structures is an array of a's where the number of a's is the number you get when you replace X by b in the X structure (a being the base and b being the prime entry). So for example, if you have a previous structure of the form $$X^{X^X}$$, you get an array of $$b^{b^b}$$ a's with the appropriate separators separating them. Sbiis believes that it is fundamental that this hold true no matter how far up you go, so for example a $$X \uparrow \uparrow (X+1)$$ structure has to turn into a an array of $$b \uparrow\uparrow (b+1)$$ a's. And my feeling is that this will be very, very difficult to pull off, which explains why it hasn't been defined yet.

In summary, we can define pentational arrays and above, and only in a somewhat clunky fashion, and I imagine there are many, many ways to do so. So it is hard to identify one particular version as the "real" BEAF. Anyway, my feeling is that if you are going to define a notation, you may as well define a new one, and not be tied to using Knuth arrows, which I don't think work that well. But, I know Sbiis feels differently; he believes Bowers' work is of utmost importance, and that a "real" BEAF exists, whether or not Bowers' precisely defined it. So opinions differ.

As for contacting Bowers, I did so a few years back, but I don't remember what was said exactly. My feeling was that Bowers did not feel that there was anything imprecise about his notation. I suppose one could write him and press him on this issue (like how are we supposed to write these arrays anyway?). I think Sbiis has had more contact with him than me, but Sbiis hasn't been around lately either.