User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-5529393-20130527105642/@comment-5529393-20130530170908

1: Nope, that's false, I just disproved it. All you can say is that _in some cases_  n!(2@) is one level higher than n!(1@). You can't then just magically say it's true in all cases. The key point is that [1...1,2] w/ [1] diagonolizes over the number of entries, so to go to the next level, you must _iterate_ over the number of entries in the array. This happens again and again in the fast-growing hierarchy.

What do you mean that [2] w/ [1] is at level \(\phi(\omega,0)+1\)? That's not even a valid expression in your notation. If you mean [2,1...1,2] w/ [1], nope, that's at level \(\phi(\omega,0)\). Look at the comparison between [1...1,2] w/[1] and [2,1...1,2] w/[1]:

[1,1,2]  [2,1,2]

[1,1,1,2] [2,1,1,2]

[1,1,1,1,2] [2,1,1,1,2]

[1,1,1,1,1,2] [2,1,1,1,1,2]

Each right entry is greater than the entry on the left, but less than the left entry in the next row. So the limit of both ordinal sequences is the same.

2:  Of course I didn't know that, because it's completely different from everything you've said. All right, I'll wait until you revamp your notation.