User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-5529393-20130530121945/@comment-7484840-20130530134514

Basically, it uses exactly the same rule as the psi or omega functions use to get to the TFB. And, yes, the above is far less than [1 (1) 1 (1) ... 1 (1) 1]. It is actually only just bigger than [1(1)2]. I used this example in the analysis above. The power really comes by when the array inside the type 2 brackets gets larger, because when it is only [21], it is the equivalent of Omega. [2k] does precisely the same thing in my notation as Omega+(k-1) does in the theta/psi functions. [21,2] = [2[21]], which, as the first entry in an array is the equivalent of adding ordinals, is equivalent to Omega+Omega = Omega*2. This continues with [21,k] = Omega*k, and therefore 21,[21] = alpha |--> 21,alpha (probably an abuse of notation, but I assume you can tell what it means), which = Omega*Omega = Omega^2, an so on. So, yes I am sort of saying that nesting the next lower number n times reaches the TFB ordinal, but only if the arrays work like mine do, because it is exactly the same rule that takes the TFB ordinal to the level of the TFB ordinal. If you think my notation is too simple, just look at the Buchholz Hydra.