# Are we allowed to compare infinities?

I’m in middle school and had a question (my dad is helping me with formatting).

We’re learning about infinity in math class and there are a lot of problems like how it’s not a number and how if you add one to infinity it doesn’t change value.

But can you have one infinity be more than another? There are an infinite amount of odd numbers and an infinite amount of even numbers, so are there the same number of odd and even numbers?

I think so, because for every odd number $n$ there is an even number $n+1$. So the odd numbers are $1,3,5,7,\ldots$ while the even numbers are $2,4,6,8,\ldots$, and as long as you stop counting at an even number the two lists will have the same number of numbers.

But there are also an infinite amount of multiples of $2$ and an infinite amount of multiples of $3$, but I don’t think there are the same amount of both. The multiples of $2$ are $2,4,6,8,\ldots$ while the multiples of $3$ are $3,6,9,12,\ldots$ So, no matter which number you stop at, the multiples of $2$ will have more numbers.

(Side question (this is dad speaking, now): is there an easy way to explain why we need to put dollar signs around mathematical expressions to make them look prettier? My daughter doesn’t know what $\LaTeX$ is, but I want to give her an explanation that isn’t horribly hand-wavy.)

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 $$thiswhichisreallyhardtoreadthis which is really hard to read$$ (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 $$nn$$ of the elements of the collection, then your list will be missing an element.

For example, with the integers, if I have $$nn$$ integers in a list, I can find the biggest one. If that’s $$NN$$, then $$N+1N+1$$ 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 $$1,2,…,n1,2,\dots,n$$. If I can pair off each element of my collection with exactly one of the integers $$1,2,…,n1,2,\dots,n$$, then my collection and $$1,2,…,n1,2,\dots,n$$ have the same number of elements, namely $$nn$$.

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 $$nn$$ with $$nn$$. But I can pair off each even positive integer with a positive integer in a different way: I can match $$nn$$ with $$n/2n/2$$. 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.