Does cardinality really have something to do with the number of elements in a infinite set?

Solution 1:

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.

Solution 2:

What is a number? It is an informal notion of a measurement of size. This size can be discrete, like the integers, or a ratio, or length (like the real numbers) and so on.

Cardinal numbers, and the notion of cardinality, can be seen as a very good notion for the size of sets.

One can talk about other ways of describing the size of an infinite set. But cardinality is a very good notion because it doesn't require additional structure to be put on the set. For example, it's very easy to see how to define a bijection between $\Bbb N$ and $\Bbb Z$, but as ordered sets these are nothing alike. Cardinality allows us to discard that structure.

Once accepting this as a reasonable notion for the size of a set, we can now say that the number of elements a set has is its cardinality.

Related threads:

1. Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
2. Cardinality != Density?
3. Comparing the sizes of countable infinite sets
4. Why the principle of counting does not match with our common sense
5. Why do the rationals, integers and naturals all have the same cardinality?

Solution 3:

Cardinality is one way to measure the size of a set, and depending on what you are interested in, it can be a very good measure. In particular, if you are interested in the null structre (and assume the Axiom of Choice) we arrive at a very natural concept of size: any two sets can be compared, and "is bigger" is a monotone and transitive relation. Furthermore this any two "isomorphic" structures will have the same cardinality.

There are other (abstract) notions of measuring the size of sets that can be used.

• The Lebesgue measure (on the reals and associated sets) is one common example. In this sense there are also more numbers in $[0,1]$ than there are natural numbers.
• We can talk about the (asymptotic upper) density of sets of natural numbers: $$\limsup_{n \rightarrow \infty} \frac{A \cap \{ 0 , \ldots , n-1 \}}{n}$$ and this will give a decidedly different measure on the size of sets of naturals.
• More generally we can pick a filter of subsets of some fixed set $X$, and then call a set "large" if it belongs to this filter, and "small" if its complement does. (In this case it is usually better to begin with a nonprincipal filter.) This sort of measuring doesn't usually result in any quantitative outputs.

But the use of comparatives such as "more" or "less" in the case of cardinality betrays the fact that mathematics (even set theory!) is done by humans, and humans have a tendency to make analogies to the real world. We have an intuitive sense of what more means, and this intuitive sense is not contradicted by the notion of cardinality. Even more, when restricted to the "finite" case, this notion completely matches up to our intuitions. While it is often dangerous to extend the more familiar finite into the infinite, in the case of analogies that can be rigourously defended, I will make an exception.