(09/03/2022, 06:36 AM)Catullus Wrote: What would happen if you used did something similar to the uniqueness criteria for tetration I proposed, but for defining non whole number rank functions in the Fast-growing hierarchy?
Catullus, this question is a question very similarly asked by John H. Conway. Conway was huge on number puzzles/game theory/weird integer iteration stuff; but what gets overlooked is his work on Hyper Operators. It is where we get Conway Chained Arrows notation: \(a \to b \to n = a \uparrow^{n} b\). And where he describes a formal algebra, very well mind you, using this notation. John H. Conway also went on to develop surreal numbers. The idea of a surreal number system, is very similar to the Fast growing hierarchy. It is intended to act in a similar manner. The thing is, these things are not proven to be equivalent (As I remember, unless there's been an update).
So you can, in the surreal number system, theoretically take a surreal number \(\omega\) and write \(2 \to 2 \to \omega = 2\). Whether the \(\omega\) in surreal numbers equates to something like \(\aleph_0\)--in a meaningful manner, is still up for grabs. I highly suggest you study Conway in this direction. But, again, this has little to do with the calculus you'll find on this forum. This forum leans heavily into calculus; and what you are asking are foundational logic questions. Which, on similar forums you might find more interesting takes.
Regards
