I've seen some videos and read some texts (non-rigorous ones) that explained the concept of cardinality, and sometimes I see someone asking if there are more numbers between the reals in $[0,1]$ then numbers in the set of natural numbers. I've read about uncountability in several places and it seems that:
The interpretation that there are more numbers is vague and inaccurate;
That the real point is about the diagonalization of the members of a set, if such a task is possible or no;
And that the idea of having more numbers is a kind of simplification that they give to the laymen;
So does cardinality really have connections with the notion of quantity of elements of a set or not? I know (I guess) that the cardinality of a finite set is the counting of the number of elements on it and perhaps this notion has leaked somehow to the notion of cardinality on infinite sets which seems to be a very different idea.
I feel that the idea of the quantity of elements of an infinite set is weird per se, when I read about the diagonalization, it made a lot more sense than the idea of quantity of an infinite set.