TITLE: Why do we distinguish between infinite cardinalities but not between infinite values?
QUESTION [7 upvotes]: More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$?
Can we distinguish between "countable divergence" and "uncountable divergence"?
I apologize in advance for the "rather naive" question.

REPLY [8 votes]: Let me just add to Kyle's answer that in cardinal arithmetic you do have a well-defined notion of uncountable sums. Because you're not trying to calculate a real number, rather you're looking for a cardinal.
If you are looking at the sum $\sum_{r\in\Bbb R}1$ as a real-valued sum, then it is indeed a strange thing to write. But looking it at as a cardinal sum, this makes perfect sense, and the cardinal is exactly $2^{\aleph_0}$. Similarly, of course, $\sum_{n\in\Bbb N}1=\aleph_0$ as a cardinal sum, and it's just a divergent series when looking at it as a real-valued sum.
So context is key.