User blog comment:King2218/FGH Things/@comment-5029411-20140304234740/@comment-5529393-20140305085221

Before you get to that, you should probably start with a concrete definition of your original function. FB100Z provided a good definition of infinite functional iteration, but how do you define [math]f_a (x)[/math] when a is a limit ordinal? You can't use the standard definition from the fast-growing hierarchy, which is

[math]f_a (n) = f_{a[n]} (n)[/math]

This doesn't work because fundamental sequences for countable limit ordinals have length [math]\omega[/math], so a[n] is only defined when n is a finite ordinal. So we need a new definition here; until we do, extending the notation out to the stratosphere is a house of cards.