User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852/@comment-5529393-20130417064430

Hmm, it doesn't seem that way to me. Each entry in the second row seems to have the same effect as two entries in the first row; for instance

n![1,1,1,1,2] = f_epsilon_0(n)

n![1,1,1;2] = f_epsilon_0(n)

The second satisfies

n![1,1,1;2] = n![1,1[1,1[1,1[...[1,1;1]...];1];1];1]

The first satisfies

n![1,1,1,1,2] = n![1,1,1,[1,1,1,1,1],2] = n![1,1,1,n]

and this basically "nests" n times in the third entry, as

n![1,1,1,2] = n![1,1,n]

n![1,1,1,3] = n![1,1,n ,2]

n![1,1,1,4] = n![1, 1, n, 3]

and so on. So both expressions have similar strength.

Essentially, the problem is that nesting is nothing new - you use it all the time in linear arrays to obtain the strength of phi_omega(0), even though it's not in the definition, it follows from iterating the reduction rules.