User blog comment:Vel!/1+w/@comment-5982810-20141011202319

To address your other concern, I was already careful to note that treating the "indexes" in my variation as "ordinals" is entirely superfluous, and is in fact misleading. Note that the fact that an ordinal is the set of smaller ordinals plays no role in either my variation of FGH, nor in the Weiner hierarchy you have presented. Instead it's simply a system by which one index is replaced with another. Without already having a familiarity of the standard  ordinal notations (which are just a convention), you would not necessarily know that this function is total. ie. there is no reason to assume that the fundamental sequence for each possible index is always a smaller ordinal. There is nowhere where a definition of the "correct" ordering of these indexes is provided, except implicitly by assuming that any member of a fundamental sequence is of lower order by definition.

The key thing is that no ordinal has a unique fundamental sequence. The thing that makes googology work isn't the ordinals themselves ... it's the sequences we chose for delimiters. Thus two delimiters can diagonalize over the same set of things and still be different functions. There is nothing sacred about using ordinals within a function. The ordinals are simply a handy way to describe the level of diagonalization at any given point within the system.