Let's say that we divide the region $(0,1)$ into $N$ bins of width $1/N$. Of course, it makes sense to take the limit $1/N \rightarrow 0$ in this configuration, because that's simply how we define an integral (the limit of the sum of evaluations of a function [times $1/N$] at the beginning of each bin, as $1/N \rightarrow 0$ is simply the integral of that function).
What I'm wondering is, does it make sense to take the limit of the cardinality of the set of bins? That is, let $S$ be the set of bins. Does it make sense to look at the limit $\lim\limits_{1/N \rightarrow 0} |S|$?
If so, would that limit be $\aleph_0$ or $\aleph_1$? I ask because there's an intuitive reason to believe either one. If you take the limit as the width becomes zero, then you could say that there's a bin at every conceivable point in $(0,1)$, which means that the set of bins should be bijective with the set $(0,1)$ which would lead me to think that the limit is $\aleph_1$. However, throughout the entire limit, the number of bins is countable, and it's hard to imagine how a countable set can discontinuously jump to an uncountable set right as the limit is reached, which would lead me to think that the limit is $\aleph_0$.
Do does the limit make sense, and if so, what is that limit?
If you must insist on an answer it would be $\aleph_0$, but the question itself is wrong.
The reason the question is wrong is that cardinals are not real numbers. Not even finite cardinals. Yes, I am well-aware that $1+1=2$ both in the real numbers and in cardinal arithmetic. But when you leave finitary operations of finite cardinals, namely when you go to the limit, then you can easily see why the two system are different.
But even if you insist that since finite cardinals are in fact real numbers, then the answer of that limit is $\infty$ rather than $\aleph_0$ or $\aleph_1$. Because the $\aleph$ number have absolutely nothing to do with real analysis and calculus. And indeed the limit of $\lim_{n\to\infty} n=\infty$.
That been said, the cardinals themselves carry their own order topology which means we can in fact talk about limits. Since this sequence is monotone, this is the same as asking what is $\sup\{n\mid n<\aleph_0\}$, and indeed the answer would be $\aleph_0$.