User blog comment:KthulhuHimself/Plexation, TaN, and such/@comment-26428969-20151014153656/@comment-5529393-20151015083114

Fluoroantimonic Acid's comment has confused you. If a function f(n) is at level alpha in the FGH, repeating f(n) n times will always be at level alpha + 1 in the FGH. This should be clear. So if by ]n{n{n{...{n}...}n}n}n[ you mean the function f(x) = ]n {x} n[ applied n times to n, then it will definitely be at level w^2 + 1 in the FGH. You've noted that Z_{n+1} = ]Z_n {Z_n} Z_n[ repeated n times will lead to a slightly larger number than Z_{n+1} = ]n {Z_n} n[ repeated n times - but it will be less than Z_{n+1} = ]n {Z_n} n[ repeated n+1 times, so it is not at a higher level in the FGH, it's still w^2 + 1.

What Fluoroantimonic Acid was saying was that it is possible to define a nested array notation that reaches epsilon_0 in the FGH. But you have to define it the correct way. In particular, it certainly cannot be the case that ]n{n{n{...{n}...}n}n}n[ is simply ]n {n} n[ repeated n times, as repeating n times only gets you +1 in the FGH. The definition must be far more involved.