User blog comment:Googology Noob/Ordinal FGH, with an actual definition!/@comment-26428969-20151228152713/@comment-5529393-20160106022303

Fluoroantimonic Acid is correct. The problem with you explanation lies here:

" Therefore:

f_w(alpha)~phi(alpha,0)"

You cannot simply observe that the first $$\omega$$ values of f_alpha(alpha) are approximately phi(alpha,0), and then conclude that one can _uniquely_ extend this function to all ordinals! Heck, even if f_alpha(alpha) was exactly phi(alpha,0) for alpha < w, you could not claim a unique extension, since there are infinitely many ordinal functions f(alpha) such that f(alpha) = phi(alpha,0) for alpha < 0.

Now, of course, you can simply _choose_ to define f_w(alpha) = phi(alpha,0). The problem is that this definition is not some consequence of a general rule that you can apply to all limit ordinals. So long as you have some other notation handy, you can define f_alpha(beta) in terms of that other notation; the problem is that this tactic won't generate new functions, so you can't go past the previous notations that you are basing the new notation on.