User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-5529393-20130527105642/@comment-5529393-20130527171306

First, \(\theta(\omega)\) should be \(\phi(\omega)\), I incorrectly said \(\theta(\omega)\).

No, [2,1...,1,2]w/[1] is not \(\phi(\omega) + 1\), that rule no longer works once you get to \(\phi(\omega)\). Look at it this way: [1,...1,2]w/[1] is \(\phi(\omega)\) or \(\phi(n)\), and so [2,1...,2]w/[1] is \(\phi(n)+1\), but \(\phi(n) + 1\) is _not_ \(\phi(\omega) +  1\)! \(phi(\omega) +  1\) is \phi(\omega)\) applied \(\omega\) times.  Remember, each time you diagonalize at a limit step, the next successor steps are that much stronger, so you can't use the same rules for them anymore.

I don't see how your statement about arrays not always evaluating applies here. You say the w/ notation means you evaluate the arrays from right to left and apply each value to the length of the previous array, so you do evaluate the arrays. So if f(n) = [1,1...,1,2], then f(f(...f(n)...)) is indeed [1,1,...,1,2] w/ [1,1,...,1,2] w/ ... w/ [1,1,...,1,2], so [1(1)2] is indeed just \(\phi(\omega) + 1\).

Saying "if there is an omega somewhere in the FGH that corresponds to a [1] in my notation, then putting an array in place of the [1] does work just like putting a larger ordinal in place of the omega" is very dangerous when making analyses like these. You need to inspect things on an individual basis.