Bowers' arrays are one of the most well-known advanced notations in googology. However the ambiguity in the notation, particularly beyond tetrational arrays, made many of the contributors of this wiki discussing about how to solve these arrays, and how far they reach. There were many theories about these, but these weren't been confirmed by Bowers himself so how the arrays are intended to be solved remained unknown.
Eventually the contributors decided to stop dealing with these, and the higher comparisons in the Fast-growing hierarchy article got removed, as in Vel!'s words, "it is now well-known that BEAF is nonfunctional above e0".
But yesterday, I found a link on Cookiefonster's site to a forum thread where Bowers himself (under the screen-name Polyhedron Dude) giving clues on how to solve these arrays. In the posts on the thread, Bowers says that the pentational and higher arrays follow the Saibian-style ordinal theory (where \(\omega \uparrow\uparrow (\omega+1) = \varepsilon_\omega\) and \(\omega \uparrow\uparrow\uparrow \omega = \Gamma_0\)). Based on this theory, kungulus, a high-level pentational array, reaches \(\Gamma_0\) level.
With this discovery, we know what Bowers has in mind. Any thoughts? Discuss here! -- ☁ I want more clouds! ⛅ 13:48, January 3, 2015 (UTC)
- Great discovery! Furthermore: Bowers says here that legion arrays should fall around the LVO, making Hyp cos' analysis/variant not the way Bowers intended it. Wythagoras (talk) 14:02, January 3, 2015 (UTC)
- Actually, Sbiis sent me a link to that forum page a month or so ago. Still neat though. Cookiefonster (talk) 15:27, January 3, 2015 (UTC)
I have only yesterday learned about how climbing method works for ordinals, so I don't have much intuition built about it yet, but one thing I don't really understand is - why do we expect same method to work Bowers's X structures? I don't see how the line of thought leading to result like \(\omega\uparrow\uparrow(\omega+1)=\varepsilon_\omega\) would lead us to results about strength of corresponding BEAF structure. Sbiis once on the chat said that, to some extent, there is no difference in using ordinals and X structures because both of these are well-ordered. I could agree if this was true, but Sbiis never gave an argument why X structures are well-ordered, nor even how we are ordering them (it was discussed when discussing well-definedness of BEAF, but the argument could be applied here too). LittlePeng9 (talk) 14:40, January 3, 2015 (UTC)
I think we should ask Bowers how to resolve simpler things like 2^X*(X+1) & 3 or even 1+X & 3. Ikosarakt1 (talk ^ contribs) 17:25, January 3, 2015 (UTC)
- Hm, that's interesting as well, but not quite as "of interest" as things like hints on pentational arrays. Cookiefonster (talk) 21:55, January 3, 2015 (UTC)
- also, vell said on the irc:
- i smacked my head when Triacontaract [that's Ikosarakt1] suggested contacting bowers
- why didnt we do that all along
- thats what we do for every other notation that people post to the wiki
- Cookiefonster (talk) 21:58, January 3, 2015 (UTC)
In addition to what CF quoted, as nitpicky as this sounds, the fact that these clarifications are not on bowers website makes me much less inclined to take them seriously -- vel! 19:19, January 5, 2015 (UTC)
Interesting find. When people started analyzing Bowers' notation, there was no evidence that he was aware of the ordinal-based hierarchies. --Ixfd64 (talk) 18:19, January 7, 2015 (UTC)