How to Count Infinity

Today we're going to count infinity. Now counting may seem elementary, like, when we say that we have five sheep, what we mean is that we have one sheep for every number from one to five. And ten sheep means one for every number from one to ten… or two to eleven. So we say that two sets have the same number of things in them, simply if you can draw a line relating every item in one set to something in the other, and vice versa, exactly once. They're partners! It's the same when we say that two plus one equals three, or three doesn't equal four: we're just describing the lines you draw to relate one set of things to another. But either way, counting sheep is boring, that is, unless you want to count INFINITELY many sheep. Like, if you had a sheep for every number between 0 and 2, would that be more sheep than if you had one for every number between 0 and 1? Nope! Because you can relate every number between 0 and 1 to its double, giving you every number between 0 and 2 (and if you want to "undo," you can just divide every number between 0 and 2 in half to get back all the numbers between 0 and 1). But there are more real numbers between 0 and 1 than there are in the infinite set of integers 1, 2, 3, 4, and so on. How on earth do we know that? Just draw some lines. For "1", draw a line to a number between 0 and 1. And for "2", draw a line to another number between zero and one. For "3", draw a line to a number between... zero and one. And so on. BUT, no matter what numbers between 0 and 1 that we've drawn lines to, we can always write down a number between 0 and 1 that disagrees with the first digit here, and the second digit here, and the third digit here, and so on… so this new number will be different from ALL of the other numbers we've drawn lines to. But we've already drawn a line from every integer, so there's no one left to be this number's partner! What's more, because of the clever way we built it, we can find an extra, lonely number like this no matter what other numbers we picked, which means we can NEVER draw lines from the integers to all of the numbers between 0 and 1 with only one line per integer… And this means that there really are more real numbers between 0 and 1 than there are in the infinite set of counting numbers 1, 2, 3, 4, and so on forever. So, Hazel Grace, some infinities truly are bigger than other infinities.