User blog comment:Tetramur/Pentational arrays and beyond - comparisons/@comment-37993808-20200108130039/@comment-39541634-20200108150947

The fact that Bowers believes BEAF to work in accordance to the climbing method is undisputable. He said it several times. The question is whether he truly understands what this claim means... and I suspect that he doesn't.

Remember that Bowers is not a mathematician. The whole point of BEAF was to create an powerful googological notation that can be intuitively understood in geometrical terms. Bowers also has a special gift for visualizing higher dimensional spaces, so he incoorperated that concept as well into his notation.

As we all know, this works perfectly well up to tetrational arrays. Then we hit our first fixed-point, and things go hay-wire when we try the usual "common sensical" extensions. So what happens after that, is anybody's guess. I'm quite sure that Bowers has some kind of intuitive picture in his mind for pentational arrays as well, but I don't think anybody (except Bowers himself) knows what that picture is.

So the question here is: Does Bowers have a good enough understanding of ordinals to properly translate the image in his mind into the world of ordinals? Can he really distinguish between a "picture" of zeta-0 ({X,X,3}&X with no climbing method) and a "picture" of gamma-0 ({X,X,3}&X using the climbing method)? Maybe he can. But it's also possible that he only has a vague anchor in the ordinal world, and he simply chose - subconsciously - the interpertation that grows faster, while being completely oblivious of what that choice implies.

As for the question of whether 3&3&3 should still have 3^^^3 entries: On his site it clearly states that it should. But that's an old site that is hardly ever updated, so it is concievable that Bowers changed his mind and didn't get around to telling us about it. The best way of knowing for sure, of-course, is to simply ask him.