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:
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The interpretation that there are more numbers is vague and inaccurate;
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That the real point is about the diagonalization of the members of a set, if such a task is possible or no;
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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.
Best Answer
The "number of elements of an infinite set" does not mean anything unless and until you choose to define a meaning for it.
So in this sense you're right: the cardinality of an infinite set does not have anything to do with the number of elements in it -- for the trivial reason that "the number of elements in it" is meaningless!
Cardinality is, however, an interesting and useful concept in its own right -- for example, to prove that transcendental real numbers exist, it is much easier to use a cardinality argument than it is to show that a particular number is transcendental, and such transfinite counting arguments are useful in many places.
It should also be clear that cardinality generalizes the usual notion of "how many elements" from finite sets, in the sense that it coincides with it when the sets are finite, and the definition does not contain anything that's obviously a trick to sneak in a different behavior for infinite sets.
Since there seems to be no other proposed generalization of "how many elements in this set" that looks like it is as useful as cardinality (this is not a deductive fact, just the practical experience that nobody has seemed to propose one, at least not while convincing very many mathematicians that his proposal was useful), it has become customary to speak about cardinality with the usual wordings of "how many" and "fewer" and "more", etc. But this is at its root merely a practical convention -- it does not purport to rest on any objective Platonic concept of how-many-ness that necessarily generalizes to infinite sets in this and only this way.
Strictly speaking, measure theory and various topological do offer competing notions of "more points" and "fewer points" that are both useful and different from cardinality. They just haven't won the practical battle for the meaning of the well-known phrasings from the finite case -- possibly because these notions requires that the sets they're used about come with more structure than just "being an infinite set", which is enough to speak about cardinality.