Help understanding countable and uncountable infinities

Question

just had some questions about countable and uncountable infinities.

1. If we take a limit that results in $\frac{ \infty }{0}$, we typically conclude that the limit is just $\infty$, correct? But if the limit results in $\frac{\infty \ \text{(countable)} }{0}$, do we say that it results in a countable or uncountable infinity? I imagine that the dividing by $0$ makes it uncountable but I wouldn’t know how to prove it either way.

2. If we take the limit of a sum to infinity, and the sum diverges, would that infinity be countable or uncountable? If it’s dependent on the sum, how would I be able to tell? Is there anything I can look for to tell?

Thanks all!

Answer 1

As mihaild points out, you are mixing up two very different concepts. They are related only by the fact that they can both be thought of as being greater than all natural numbers (and thus all real numbers), and so we call them both "infinite". But that is pretty much the end of any relationship between the two concepts.

• $\infty$ and $-\infty$ are about topology. They are two additional points we stick at each end of the number line to make it compact - that is, to make it look like a closed interval $[a,b]$. If we have a sequence $\{x_n\}_{n\in\Bbb N} \subseteq [a,b]$, we are guaranteed that it has limit points (there may be many of them, not just one - for example $\{(-1)^n\}_{n\in\Bbb N}$ has two limit points). But $\Bbb R$ is not the same. There are sequences in it without limit points, such as $\{n\}_{n\in\Bbb N}$. This can be rectified by just adding two additional points to $\Bbb R$. These points allow us to talk about limits in these directions that otherwise would be undefined.
• "countable", "uncountable", $\aleph_0$, $\beth_1$, etc, are not about topology at all. They are about counting. A set us countable if it can be put in 1-1 correspondence with a subset of the natural numbers. The cardinality of one set is at most as high as another if it can be put in 1-1 correspondence with a subset of the other.

Note that the first concept involves limits, while the second involves correspondences. They are just two different things. There is no concept of limits at cardinal infinities, and there is no concept of a correspondence with $\infty$.

Because they both can be thought of as sitting "just" beyond the natural numbers, it seems natural to identify $\infty$ with the smallest infinite cardinal $\aleph_0$. But even if you do, that identification doesn't somehow tie all the rest of the cardinals into the concept of limits.