User blog comment:Deedlit11/Is BEAF well-defined?/@comment-25418284-20121121205547/@comment-5529393-20121201031625

@Ikosarakt1  Very interesting findings! Unfortunately I am not in a position yet to verify your comparisons;  I have to study legions more. But it would be very interesting if BEAF only went up to the Ackermann ordinal.

@FB100Z  Yes, we can go farther! The easiest way to define it would be to ignore the prime block, and simply define a copilot location for every pilot location. If the pilot is located at entry X+1, then the copilot will be at location X. Otherwise, the pilot is located at some limit ordinal X, and the copilot will be located at X[p], the pth member of X's fundamental sequence, where p is the prime entry. So decrement the pilot, replace the copilot with the array with the prime entry decremented by one, and leave everything else the same.

If you would rather have more passengers, you can do the following:  Just about every ordinal notation notates ordinals by starting from some base ordinals and applying certain functions to it. For example, the standard notation for the Bachmann-Howard ordinal has base ordinals 0 and Omega, and uses the functions a,b -> a + b, a -> omega^a, and a -> psi(a). So we can define a norm function as follows:

N(0) = N(Omega) = 0

N(a + b) = N(a) + N(b) + 1

N(omega^a) = N(a) + 1

N(psi(a)) = N(a) + 1

If an ordinal has more than one notation, its norm is the smallest of the norms of its notations.

Then, given a pilot at entry X, we can define the passengers of the array as those ordinals less than X that have norm at most p. This is a finite number of passengers, since there are only finitely many notations with norm at most p. We let the copilot be the largest of the passengers.

So there are some workable ways that we could turn Bowers' BEAF into an ordinal notation. I am still interested in just how far Bowers' legions go compared to ordinals, unless Ikosarakt1 has cracked that already.