The sizes of infinity
Last week a student visited the Drop-In Centre to talk about the different sizes of infinity. His lecturer had been talking about the sizes of sets and had made an off-hand comment that there were different sizes of infinite sets, and he wanted to know what the hell that meant.
So I explained it. It's not the simplest explanation, because you have to define some new ideas in order for it to make sense. But it is one of my favourite things about the families of number that some are the same size and some are different sizes, even though they are all infinite.
So I want to explain it again here...
The key to the whole thing is to realise that when talking about infinite sets, that counting is not really a useful way to find the sizes of sets. When you have a lovely finite set, such as the letters in the word "sluiced", you can simply count the members of the set and you know how many there are (7 in this case). But the process of counting works because at some point you stop, and the number you're up to is the number of objects there are.
But what if you can't stop? If your set has infinitely many objects in it, you'll never get to a final number where you can stop and say "this is how many". We need a new way to talk about the sizes of sets, and the way that we use is in some sense even more fundamental than counting.
If you had two piles of things and were asked which pile has more, you'd probably count both and whichever had the bigger number would be bigger. But there's another way: you could pair them off, one object in each pile, and if one pile runs out before the other one does, then you know that the other pile must be bigger. And if you are able to pair off everything in both piles, you k