I was equally annoyed by this part. You are very generous with your very close to making an incorrect statement. I would straight away say it’s both misleading and wrong.
>no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes
What is misleading or wrong about that? What specific statement is incorrect?
>this doesn’t mean that the sets of real numbers and natural numbers have different sizes
I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does).
> But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers.
> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation.
So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.