Forum:Word of God about pentational arrays and beyond

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 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 his mind. Any thoughts? Discuss here! -- ☁ I want more clouds! ⛅ 13:48, January 3, 2015 (UTC)