User blog comment:Eners49/A whole new superclass of infinities?/@comment-30754445-20180722034443

A few points to consider:

1. "Absolute infinity" is concept that makes no coherent mathematical sense. Proof: think what will happen when you add 1 to "absolute infinity". :-)

2. Even if it did make coherent mathematical sense, you wouldn't be able to "extend" aboslute infinity in any way. After all, that's what "absolute" means.

3. The limit you're probably looking for here, is the smallest "uncountable oridnal", also known as ω1. This is the absolute limit of the fast growing hierarchy, because ω1 does not have a fundamental sequence.

So if you want to extend the FGH to (non-finite) ordinal output, fω 1 (n) is a good place to begin your extension.

4. Defining fω 1 (n)=ω for all n is, of-course, useless. We a function that actually grows. Not sure how to do this in an efficient way, though.

5. If you try to continue in this manner, you're going to need to define the FGH for ordinal input as well. For example, according to the usual rules of the FGH:

fω 1+1 (3)=fω 1 (fω 1 (fω 1 (3)))

But fω 1 (3) is a (non-finite) ordinal! So we need a way to evaluate things like fω 1 (ω).

6. The entire concept of Ordinal Collapsing Functions (those ψ's that you keep seeing popping around here) is somwhat similar to your idea. The

The generation rules are different (the ψ's aren't based on the FGH) but the general idea is the same: The ψ function uses ω1(usually notated as a non-red Ω) to create large countable ordinals, just like your proposed fΩ function. In fact, it is customary to use multiple ψ functions (ψΩ, ψΩ(2), ψΩ(3), ... ) in a way that closely mirrors your suggestion. Defining ψΩ(n+1)(0) = Ω(n) actually makes tons of sense (though the standard practice is to define ψΩ(n+1)(0) as the slightly larger ordinal εΩ(n)+1). By the way, if we're using collapsing function in this way (and doing it properly) then the absolute limit of the entire system would be ψ(ψΩ(Ω(Ω(Ω(Ω(...(0)...)))))(0)) = ψ(ψI(0)), which is - to be honest - f***-ing huge. Not sure if an FGH-based extension can reach the same level, though. My guess is that it can, if we are very very very careful with our definitions (when doing these things, even a tiny diviation from the optimal path would result in a much weaker system)

(

So if you like, you could say

Recall

, which is pro

.The smallest of these is ω1 (not to be confused with the Church-Kleene Ordinal,

ck, which is smaller).  This ω1 is the absolute limit of what you can plug into the FGH, because it has no fundamental sequence.