What is the point of studying the cardinality of infinite sets?

Question

Is it just for the sake of completeness in the understanding of cardinality of sets in general?

Or is there any particular uses of such knowledge apart from further abstractions of the concept?

Thank you!

Answer 1

"Studying the cardinality of sets" means "studying when there is or is not a bijection, injection, or surjection from one set to another" and it is one useful way of telling sets apart.

One early illustration of this is due to Georg Cantor, who used cardinality to give a simpler proof of something already known, the existence of transcendental numbers. Namely, he showed that there is a surjection from the set $\mathbb N$ of natural numbers to the set $A$ of algebraic numbers, but there is no surjection from $\mathbb N$ to the set $\mathbb R$ of all real numbers; whence it follows that $A\ne\mathbb R,$ i.e., there are real numbers which are not algebraic.

Answer 2

If I told you that every set of reals is Borel, how would you prove or disprove this?

Once you study cardinalities, you learn the Borel sets have $\aleph_1$ "levels", each level with $2^{\aleph_0}$ sets; therefore there exactly $2^{\aleph_0}$ sets of reals which are Borel sets. Being an educated man, you know of Cantor's theorem and you know there are $2^{2^{\aleph_0}}$ sets of reals. So definitely not all the sets of reals are Borel sets.

Well, what about Lebesgue measurable sets? Maybe not all sets of reals are Borel; but are all the sets which are Lebesgue measurable Borel?

Again, the answer is no, because of a simple counting argument: the Cantor set is Borel, and null. So every subset of the Cantor set is Lebesgue measurable. Again, there are only $2^{\aleph_0}$ Borel sets, but there are $2^{2^{\aleph_0}}$ subsets of the Cantor set.

You can do these games again with functions which are continuous (or rather close to being continuous, i.e., Borel measurable). It tells you that in the grand scheme of mathematical objects, the ones we care about are usually "the pathological exception" and not the other way around.

Answer 3

To play the Devil's advocate: distinguishing between different infinite cardinalities is in some senses not as essential as one might expect, and in those senses the question is more than justified. Apologies in advance as I'm no expert in proof theory, but I'll try to give the idea.

As an example, many results are classically stated and then proven assuming something like the axiom of powerset, which says that the set of all subsets of an infinite set must exist, and then we know by Cantor's theorem that this set of subsets must have a strictly larger cardinality than the original infinite set. But in fact, these results can often be restated and proven in a form which assumes nothing of the sort and which, while strictly speaking weaker, is in fact adequate for whatever situations we are likely to care about.

The sort of weaker axiom we might use instead will say that all subsets of a particular type exist, and the subsets of that type will turn out to be equinumerous with the original infinite set, so distinctions of cardinality are not really relevant. Such axioms belong to restricted subsystems of second-order arithmetic, which only really talks about infinite sets which are countable.

In fact, even the very strong large cardinal axioms of set theory have much weaker "miniature" versions which merely state that a certain "recursively large" countable ordinal exists, and such axioms suffice to prove all the consequences that a non-logician is likely to care about.

All of this is very valuable to anyone who wants to know what's needed to proven a given result: in other words, which axioms are we actually required to use when proving the result, and which ones are just overkill? This is the programme of reverse mathematics, and it is especially relevant if you harbour doubts that the stronger axioms are even consistent (and the stronger your axioms are, the more cause you have to be concerned about this).

Answer 4

First of all, I think that giving the distinction between "real" and rational numbers as the reason or point for studying cardinalities, or even as an example is, well, questionable. Undergraduate math glosses over problems surrounding existence of "real" numbers, but even if the question of their existence was definitively answered once and for all, this is still not a good example.

The reason this is not a good example is that mathematics doesn't try to answer "why" question, it's the job of philosophy to answer that sort of questions. I mean, answering that we study cardinalities of infinite sets for the benefit of distinguishing some sets immediately begs the question: why do we need to distinguish between those sets?

Now, the question of purpose is, itself, questionable, but since you were wondering what's the purpose of something, I think it is safe to assume you believe that, at least man-made objects are conceptually capable of having purpose.

I'm not sure there's a single purpose for this study, however one important aspect that I can see is that this study would enable humans to conceptualize about continuum. We have intuitive notions of what continuum is: objects from macrocosm often appear to occupy continuous patches of space, the hands of a clock aren't supposed to skip over any division on the clock's face, no matter how fine etc. But these naive notions of continuum may lead us to paradoxes, where we would not be able to decide on the outcome which best reflects our intuition, like, for example https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox . Studying the cardinalities of infinite sets thus allows to resolve these paradoxes, or, at least to hope to one day to be able to :)

Answer 5

First, we say that two sets have the same cardinality if there is a 1-1 mapping between their elements.

Then, we can show that there is no 1-1 mapping between a set and the set of all its subsets (called its power set). This means that there are an infinite number of possible cardinalities, each gotten by successively taking power sets.

In particular, since the integers are countable (by definition), the set of all subsets of the integers, which can be shown to be equivalent to the reals, is not countable.