User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-28606698-20170415185346/@comment-30754445-20170416103854

The problem is that there's nothing in Pellucidar's description that even hints at a nonrecursive process below ωck. He jumps straight from "enumerating fixed points" (which is definitely a recursive process) to admissable ordinals.

For such a notation to work, there needs to be a "bridge" entry which bridges the gap between the recursive ordinals and the admissables. For example, we could use the 4th number in the array to enumerate some (uncomputable) fundamental sequence of ωck. Basically we could say that the 4th number tells us which recursive system we will be using (assuming the 5th number is 0).

Of-course, for this to work, we'll need much more space than a simple triad of numbers {a,b,c} to enumerate all the ordinals in the Nth system... I suppose we could allow a,b,c to be ordinals themselves, but that will only delay the inevitable. Given that these first 3 entries are called upon to describe an arbiterarly strong recursive system, I don't see how the whole thing could work.