User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-39541634-20190722132933

@P進大好きbot

It's still a huge gap. At the very least, it would be appropriate to spread the scale further apart at that point and leave the intermediate level as "To Be Determined".

But really, we don't even need to do that, because we do know what comes between them: The PTOs of various theories. And not having the fundamental sequences at hand (yet!) is not synonymous to complete ignorance. We still know enough about these levels to meaningfully talk about them!

In fact, we can even define explicit numbers as boundaries, without fundamental sequences:

For any theory X, we can define fPTO(X)(n) as follows:

(1) Let us look at all the recursive functions from N to N which can be proven total in theory X using n characters or less (we'll need to give a specific formulation and syntax for theory X for this to be a regorous definition, of-course).

(2) If no such function exist, return 0.

(3) Otherwise, among these function, pick the function that gives the maximum value when we plug n into it, and call it F.

(4) fPTO(X)(n)=F(n).

Of-course, ideally we would like the entire computable ruler to be defined with the good old FGH and with fundamental sequences. But the above can serve as a temporary measure and a rough guide of scale. After all, nobody seriously believes that the strength of (say) fPTO(Z2)(n) is going to change substantially when we switch systems.