That's fun but it's not clear how interesting those numbers are compared to real numbers, which have turned out to be pretty interesting over the years

>What if you have 0.9̅4?

Well, you fundamentally can't. If the 9s go on for forever then you never reach a point where you can add the 4. The definition of infinity precludes anything after infinity, because it never ends so you can never get there.

In this alternate number system you can think of it as a tuple: (.9̅, 4). Which is larger than (.9̅, 3) and smaller than (.9̅, 5).

That is, until Cantor showed otherwise

Are you saying that Cantor showed you can have 0.9..4?

I'm not even remotely an expert here, so I might certainly be wrong, but I don't understand how Cantor's theorems show an ability to stick a finite number and stop on the end of an infinite number.

Yes, one thing Cantor showed is that it is somehow meaningful to define numbers like infinity+1, where there are an ordered infinity of elements followed by one more element. Sets like this are called ordinals

https://en.wikipedia.org/wiki/Ordinal_number

So you could, if you like, define 0.9...4 to be a bunch of digits indexed by the ordinal ω+1. However the thing you have now defined isn't really a representation of a real number any more, unless you just ignore all the bits after the ... I guess.

Huh, interesting. Looks like I've got some more reading to do.

Thank you!