User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852

Okay, I've looked at your notation, and unfortunately you've misgauged the comparison between it and the FGH. Your reduction rule

n![1,...,1,1,k,@] =  n![1,...,1, [1,...,1,1,1,@] ,k-1,@]

is actually weaker than Bowers' rule, which is

{a, b, 1, 1, 1, ... 1, c #} = {a, a, a, ..., {a, b-1, 1, 1, ..., 1, c #}, c-1 #}

Note that the second to last entry of the right hand side is another expression ending in c #, so you can reduce it again to an expression ending in c-1 # using the same rule, and reduce the second to last entry of that using the same rule again, so you wind up with b nested expressions ending in c-1 #. So his rule is certainly stronger than yours. Nevertheless, his notation for linear arrays is still at the level omega^omega in the FGH, same as if he had defined

{a, b, 1, 1, 1, ... 1, c #} = {a, a, a, ..., b, c-1 #}.

So your notation, with a weaker reduction rule, only reaches omega^omega in the FGH as well. Because it is weaker it needs one more variable to achieve the same strength as Bowers' arrays, but the limit is still omega^omega. The reason neither of your notations goes any farther is that these stronger reduction rules at limit ordinals are doing the same thing as the reduction rules at successor ordinals, so all they achieve is to map to the ordinal + 1, which is no big deal. For example, {a, b, c, d} using Bowers' normal rules is about the same as {a, b, c+1, d} using the weaker rule listed above, but that makes no difference in the long run.

So where does your analysis go wrong? The first mistake is at

n![[1,1,2],1,2] ~ f(ω^2)+ (ω^2) (n) ~ f(w^2).2(n)

Applying a function twice does not double the level in the FGH. In fact, you have to iterate the function n times just to advance it one level in the FGH! So a correct analysis would be

n![1,1,3] = n![1,n,2] = f_{w^2 + 1}(n).

As for multidimensional arrays, your nesting rule is pretty much the same as Bowers' reduction rule, so you get the same strength, which is omega^omega^omega in the FGH.