User blog comment:Scorcher007/The whole googology in one diagram/@comment-4224897-20180323140711/@comment-30754445-20180323143637

I'm sure what you're objection is. Where, in the diagram, do you see nonrecursive ordinals plugged into the FGH?

As a side note, it isn't true that plugging nonrecursive ordinals into the FGH is a-priori meaningless. They still have countable fundamental sequences. The problem is that these sequences will usually be uncomputable, and therefore of limited use.

For example, you can define fωck(n) = BB(n). From there we can prceed normally and recursively all the way up to (and not including) fωck×2(n).

We can then define fωck×2(n) with the following Busy Beaver variant: "The largest number of ones printed by a turing machine with n states + an oracle for the function BB(n)"

Then we can define a series of such oracles and notate them with ordinals. This can be done recursively up to (and not including) fωck²(n).

And so on.

The definitions will naturally get quite unwieldy, qute quickly. Worse: actually calculating the outputs for these functions would be impossible in general. But these are, nevertheless, well-defined functions (and special cases can - in principle - be calculated on a case-by-case basis).