Are we allowed to compare infinities?

Solution 1:

In response to the side question: it alerts the system that you want them to be rendered as symbols rather than left alone. You don't want the system to try to render everything as math because then it looks like $this which is really hard to read$ (unless you force it to insert spaces etc. by hand). In this question it wasn't really all that necessary.

As for the question: there are two main kinds of infinity that come up frequently in mathematics. The one that is mentioned here is called infinite cardinality. This means you have some collection of things, and there are infinitely many things in it. We say a collection is infinite in mathematics if whenever you list any number $n$ of the elements of the collection, then your list will be missing an element.

For example, with the integers, if I have $n$ integers in a list, I can find the biggest one. If that's $N$, then $N+1$ is an integer not on the list. So there are infinitely many integers (as you know).

The idea being used here is pairing off elements of your collection with the integers $1,2,\dots,n$. If I can pair off each element of my collection with exactly one of the integers $1,2,\dots,n$, then my collection and $1,2,\dots,n$ have the same number of elements, namely $n$.

We define infinite cardinality in the same way: two infinite collections have the same cardinality ("number of elements") if I can pair off each element of one with exactly one element of the other. The tricky thing with infinite collections is this word "can". You might think that there are fewer even positive integers than there are positive integers, because all even positive integers are positive integers but odd positive integers exist. What you've done is paired off each even positive integer with a positive integer in one way, by matching $n$ with $n$. But I can pair off each even positive integer with a positive integer in a different way: I can match $n$ with $n/2$. Then I've actually paired off each positive integer with each even positive integer. So they have the same "size", at least if we decide to define size this way.

A surprising fact discovered by Georg Cantor in the 1800s is that not all infinite collections have the same cardinality. The most familiar infinite collections with different cardinalities are the integers and the real numbers. There are more real numbers than there are integers.

The way that Cantor showed this is basically the same way that we showed that there are infinitely many positive integers: pair off each positive integer with a real number (producing an infinite list of numbers) and then present a real number which isn't on the list. The hard part is again this word "can": he had to come up with a recipe for a number not on the list no matter what list he was given, so his recipe for the "missing" number has to depend on the given list in a clever way. Cantor also showed that there are infinitely many different infinite cardinalities, using basically the same idea.

Solution 2:

This is a wonderful question; unfortunately many of the answers jump directly to explaining cardinality as if that is somehow THE only valid response to your musings.

Now, there's nothing wrong with cardinality -- it is an interesting and important concept in mathematics, and often useful. But there's no rule thats says it is how you must think about the size of infinite sets.

If you need to do something that cardinalities are not well suited for, it is completely allowed to use a different concept instead. In mathematics we're always free to choose the concepts and definitions that will help us reach our goal, as long as we're clear about what we're doing (and as long as our choice of definitions doesn't change what the goal is, of course. If someone else sets a goal for you, you cannot wiggle out of that by redefining the words they used to mean something different from what they had in mind).

For example, if we're talking about sets of points in the plane, then cardinality is often too crude a way to speak about their size. A square of side length 1 cm and a square of side length 2 cm both contain infinitely many point -- and according to cardinality they both have the same infinite cardinality becuase we can match up the points one to one. But it is still reasonable to want to say that the 2cm square is $4$ times as large as the 1cm square -- so for that we choose to speak of area (or, in fancier words, "measure") instead of cardinality. And that is completely all right.

In your example, you have two different sets of natural numbers, $$ \{2,4,6,8,\ldots\} \quad\text{and}\quad \{3,6,9,12,\ldots\}$$ and cardinality says they have the same size. And sometimes the kind of size cardinality talks about is what you need, and if that is the case you go on thinking of them as equivalent.

For other purposes, though, you might need to express the intuitive fact that the first set contains half again as many numbers as the second. That's fine too; we just need to find something else than cardinality for making that intuition precise.

For this purpose we can use the limiting density that Julian Rosen speaks about: When we have a set of numbers we can ask for how many numbers are less than $k$, for any possible $k$ -- and we can then also ask about the fraction of numbers less than $k$ that are in our set, namely the fraction $$ \frac{\text{count of numbers in the set less than }k}{k} $$ Now in both of the cases $\{2,4,6,8,\ldots\}$ and $\{3,6,9,12,\ldots\}$ it happens that there's a certain number $d$ such that when $k$ is large, the fraction above is always close to $d$. (In technical terms, $d$ is called the "limit" of the fraction, and there's a precise definition for what we mean by that, but you won't have learned that yet in middle school). In that case we can call $d$ the density of numbers in the set. The density of $\{2,4,6,8,\ldots\}$ and $\{1,3,5,7,\ldots\}$ are both $\frac12$, confirming your intuition that they have the same size. But the density of $\{3,6,9,12,\ldots\}$ is only $\frac13$, so in the sense of densities there are indeed less multiples of 3 than there are even (or odd) numbers.

This concept doesn't solve everything, though. First, there are sets of numbers that don't even have a density according to this definition -- for example, the set of numbers that are written with an odd number of digits in base ten. In that case the fraction above keeps taking different values between between $\frac1{11}$ and $\frac{10}{11}$ as $k$ increases, never closing in on a single limit.

Second, sometimes we'd like to compare the sizes of sets that have the same density -- for example, the set of perfect squares and the set of prime numbers both have density $0$, but it turns out that, in an appropriate sense there are still "more" primes than there are perfect squares. Namely, the number of primes less than $k$ grows definitely faster than the number of perfect squares less than $k$, when viewed as functions of $k$. This too can be made precise if we put sufficient work into it.