Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
Suppose you have two buckets, A and B, each with a bunch of doodads in them. There are (at least) two ways you could determine which bucket has more doodads in it:
1. Count up the number of doodads in A, then count up the number of doodads in B, and then see which number is bigger.
2. Take out a pair of doodads, one from each bucket, and then set them aside. Then take out another pair and set those aside, too. Keep doing this until you run out of doodads in one of the buckets. The bucket that runs out of doodads first must have had fewer doodads in it.
Now, Method #1 will work as long as there is a finite number of doodads in one or both of the buckets. But if there are an infinite number of doodads in both buckets, you run into a bit of a problem. Which, if either, of those infinities is larger? You can’t really say. You might say, well, infinity = infinity, so they must have the same number of doodads in that case. And indeed, this was the general attitude of mathematicians until around the 1870s.
Enter [Georg Cantor](https://en.wikipedia.org/wiki/Georg_Cantor), who said, hold up, if indeed it’s true that infinity = infinity, then we should also come to the same conclusion (i.e., that both buckets have the same number of doodads) if we apply Method #2. Cantor then showed that actually sometimes if we start pairing off doodads, we’ll end up running out of doodads in one bucket before the other, even when both have an infinite number of doodads in them. Thus was born the idea of differing “sizes” of infinity.
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