How do you use transfinite cardinal numbers to describe the size of an infinite set?

Question

The size of an infinite set can be expressed by transfinite cardinal number.

I read about it in Wikipedia. But did not get how is it used to describe the size of an infinite set?

Answer 1

The question is what are numbers anyway?

If you only know about the natural numbers, then what is $\frac12$? How would you even indicate that something has size $\frac12$? You would say that it can be put exactly twice into the unit. But also four times into two units, or seventeen times into thirty-four units.

In such way you declare that $\frac12$ is a size which can be represented as a ratio between two finite sums of units. That's great! Now what about $\sqrt2$? How do you define that?

Well, again, you can be formal and argue that this is just a formal positive solution to the polynomial equation $x^2-2=0$. It doesn't truly represent something which is tangible in nature. We can make things even worse, and ask about $-e^\pi$, or something like that. But you get the idea: we use formal mathematical definitions that were derived from ultimately-naive-understandings.

But then, what is the size of a finite set? If we know about the natural numbers, that's great, we can just say that if you can match the elements of the set with some $\{0,\ldots,n-1\}$ for a natural number $n$, then the set has exactly $n$ elements. Great!

But what if we don't have the natural numbers? What if we want to work without the natural numbers because we plan to define them using sets and cardinalities? What then? Remember that you don't have to be able to count up to five in order to tell that a healthy human being has the same number of fingers on each hand. You can just match each finger on the left hand, to the corresponding finger on the right.

This means that we can say when two sets have the same size using bijections. And that describes when two sets have the same size. In set theory this gives us the ability to define the natural numbers without referring to counting principles.

Great. But bijections are not a privilege of finite sets. We can talk about bijections between any two sets: is there a bijection? Is there a bijection between one of the sets and a subset of another?

This lets us define the cardinality of a set $A$ as some formal object which represents the class of sets which have a bijection with $A$. Assuming the axiom of choice, however, we can actually turn this canonical object into a particular set in the mathematical universe: the axiom of choice tells us that every set can be well-ordered, and therefore for every $A$ there is a least ordinal in bijection with $A$, and that ordinal is the cardinal number of $A$.

For the case $A$ being the natural numbers, we call that cardinal number $\aleph_0$. We can prove that $\Bbb R$ has a strictly larger cardinality, so the cardinal number of $\Bbb R$ is larger than $\aleph_0$. But without additional hypotheses we cannot determine what is the exact value of the cardinal of the real numbers.

It is important to make two remarks at this point:

1. Cardinal numbers carry an arithmetic structure. We have summation, products and exponentiation.

   Cardinal numbers do not extend the real numbers. They are compatible, as far as finite cardinals with finitary operations are concerned, with the natural numbers. But once we start talking about infinite summations or products, the definition in the world of cardinal numbers is very different than the one in the real numbers.

2. There are other sense of transfinite numbers: ordinal numbers, surreal numbers, and more. These are not used to measure the cardinality of a set, but they have their uses. This answer, however, is already long enough.