I saw a yt vid on largest numbers etc, and extremely baffled by how many versions of infinities are. Didnt know it has premium version like absolute and the regular infinity. lol
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There are many concepts unrelated concepts that are covered by “infinity”.
There is the concept of “growing without bounds”, attached to limits and usually representes by ∞.
There is the concept of “How many things do I have?” which is attached to cardinals and represented by Aleph and Beth numbers (among others).
There is the concept of “How do I arrange/order infinitely many things?”, which builds up on cardinals (infinitely many things), and is attached to ordinals and represented by omega and epsilon numbers (among others).
Then, as we explored these, we realized wouldn’t it be nice if we could deal with these infinite numbers just like we do with normal ones, or at least as much as possible. At the same time, we were still trying to find a way to make formal Newton’s calculus and the problem of infinitely small values (ghosts of departed quantities) turns out, it can all be made quite formal and attached to the hyperreals.
Each of these kinds of infinities represent a different concepts and need to be manipulated using different rules. Because infinity is weird.
Growing without bounds doesn’t really have any subtleties. If it grows without ever stoping then that’s that. Not much else to do.
Counting (Cardinals) is different though. Basically to say we have the de number of two kinds of things, what we do, is make sure we can match them one-by-one without repeat or forgetting anyone. Sure there are infinitely many natural numbers, and adding one or even doubling the amount of numbers we have doesn’t really change anything. But there are some operations that will change that.
For example, we can’t match naturals and real numbers one-by-one. This proven by showing that if we had such a matching, we could find a real number that hasn’t been matched yet, but no natural number that hasn’t been matched, even though there are infinitely many of them.
Ordering (ordinals) is a bit more subtle. We need the one-to-one correspondence, but we also need to do it so the order is preserved. So adding an element at the end is not the same as adding at the beginning once you have infinitely many of them.
All of these are only worried about operations about the natural numbers (addition and multiplication basically) because it’s all about counting.
The hyperreals are what happens when we try to smooth out these differences but add division as well. You end up with not only infinitely big (called transfinite) numbers, but also infinitely small ones (infinitesimal).
In any case, things get weird fast when you try dealing with infinity.
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