User blog comment:Deedlit11/Is BEAF well-defined?/@comment-25418284-20121121205547/@comment-5529393-20121123004850

We could certainly do that, and it would go very far, as the ordinals with constructive notations go very far.? The construced notation would be quite similar to the fast-growing hierarchy, and would match up with the fast-growing hierarchy at all the major ordinals (epsilon_0, Gamma_0, etc.).? What I would like to know is where Bowers' legions, lugions etc. fit in the fast-growing hierarchy.

Bird's notation is very cool, and the fact that he compares his notation quite specifically to the fast-growing hierarchy is even cooler.? HIs notation goes all the way to the Bachmann-Howard ordinal.? It would be very nice if we could establish exactly where Bowers' notation fits in the fast-growing hierarchy, assuming Bowers' notation could be precisely defined.? Bird gives a precise correspondence in his paper "Bowers' Named Numbers", but I'm not sure if his analysis can be considered definitive. (He rates meameamealokka oompa as somewhat beyond the Large Veblen Ordinal, well below the Bachmann-Howard ordinal.)?

Although we don't have a formal definition of BEAF for pentational arrays and beyond, it seems intuitively natural that pentational arrays go up to Zeta_0 = phi(2,0), hexational arrays go up to phi(3,0), and n-ational arrays go up to? phi(n-3, 0).? F_phi(omega, 0) (n) would correspond to {a, b, n} & c; I guess this is what Ikosarakt1 means by "{a, b, c} ranges has level phi(w, 0)".? It seems then that F_phi(phi(omega, 0), 0) (n) would correspond to {a, b, n} & c? & c,? F_phi(phi(phi(omega, 0), 0), 0) (n) would correspond ot {a, b, n} & c & c & c, and that F_Gamma_0 (n) would correspond to n & n & n ... &n with n n's.? This is as far as I have hypothesized, as my understanding of legions is incomplete.? It would be interesting to see where the small and large Veblen ordinals fall in Bowers' notation.