User blog:Eners49/A whole new superclass of infinities?

The set of natural numbers is infinite, being {1, 2, 3, ...} extending infinitely. There are things called ordinals which are an extension of the natural numbers into the realm of infinity, and are denoted with Greek letters: \(\omega, \epsilon_0, \zeta_0, \phi(4, 0), \) etc. We can obtain some ordinals such as epsilon-zero or the Feferman-Schutte ordinal with recursion, and there are also other ordinals such as omega one of chess or the Church-Kleene ordinal which do not employ recursion.

However, just like with natural numbers, there is a limit as to how long we can extend the ordinals. The fast- and slow-growing hierarchies use ordinals. So you could have \(f_{\omega}(3), f_{\epsilon_0}(100), f_{\phi(4, 0)}(1)\), etc. which are all large, but finite, numbers. But what would the subscript in the function have to be so the function returns an infinite number???

Sbiis Saibian once created an "ordinal" called absolute infinity, stating that it was the largest possible ordinal number, and he denoted it with a red capital letter omega. Since that's tedious for me to do and Wikia doesn't let me add color to text (FUCK YOU WIKIA), I'm going to denote it differently: \(\Omega(0)\). As far as I know, no one has used the capital omega as a function, so I'm going to do that. This \(\Omega(0)\) is our absolute infinity, and we can say it is the number such that \(f_{\Omega(0)}(x) = \omega\). In this case, the \(\omega\) is the ordinal, and is an infinite number. But what should the subscript in the fast-growing hierarchy function be so that it returns absolute infinity as the value? I'm going to say that \(f_{\Omega(1)}(x) = \Omega(0)\). In general, \(f_{\Omega(a)}(x) = \Omega(a-1)\).

So of course, I could have "hyperordinals" \(\Omega(\omega)\), \(\Omega(\phi(4, 0))\), or even \(\Omega(\Omega(0))\). Of course, I could find a way to iterate that as well, but I'll stop here for now until I understand this better.

Someone who is skilled with working with ordinals and stuff, please tell me more about what I have written. Chances are I've written a bunch of shit, "hyperordinals" aren't even possible, and I'm probably going to get laughed at like SammySpore, the creator of Sam's Number. Anyway, please comment below your thoughts!