For some reason, lately I’ve been seeing a bunch of mentions of Banach Tarski. B-T is a fascinating case of both how counter-intuitive math can be, and also how profoundly people can misunderstand things.<\/p>\n
For those who aren’t familiar, Banach-Tarski refers to a topological\/measure theory paradox. There are several variations on it, all of which are equivalent.<\/p>\n
The simplest one is this: Suppose you have a sphere. You can take that sphere, and slice it into a finite<\/em> number of pieces. Then you can take those pieces, and re-assemble them so that, without any gaps, you now have two<\/em> spheres of the exact same size as the original.<\/p>\n Alternatively, it can be formulated so that you can take a sphere, slice it into a finite number of pieces, and then re-assemble those pieces into a bigger<\/em> sphere.<\/p>\n This sure as heck seems<\/em> wrong. It’s been cited as a reason to reject the axiom of choice, because the proof that you can do this relies on choice. It’s been cited by crackpots like EE Escultura as a reason for rejecting the theory of real numbers. And there are lots of attempts to explain why it works. For example, there’s one here<\/a> that tries to explain it in terms of density. There’s a very cool visualization of it here<\/a>, which tries to make sense of it by showing it in the hyperbolic plane. Personally, most of the attempts to explain it intuitively drive me crazy. One one level, intuitively, it doesn’t, and can’t<\/em> make sense. But on another level, it’s actually pretty simple. No matter how hard you try, you’re never going to make the idea of turning a finite-sized object into a larger finite-sized object make sense. But on another level, once you think about infinite sets – well, it’s no problem.<\/p>\n The thing is, when you think about it carefully, it’s not really all that odd. It’s counterintuitive, but it’s not nearly as crazy as it sounds. What you need to remember is that we’re talking about a mathematical sphere – that is, an infinite collection of points in a space with a particular set of topological and measure relations.<\/p>\n Here’s an equivalent thing, which is a bit simpler to think about:<\/p>\n Take a line segment. How many points are in it? It’s infinite. So, from that infinite set, remove an infinite set of points. How many points are left? It’s still infinite. Now you’ve got two infinite sets of the same size. So, now you can use one of the sets to create the original line segment, and you can use the second one to create a second, identical line segment.<\/p>\n Still counterintuitive, but slightly easier.<\/p>\n How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets, Now take the set of even numbers, and map it so that for any given value i, f(i) = i\/2. Now you’ve got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)\/2. Now you’ve got a second<\/em> copy of the set of natural numbers. So you’ve created two identical copies of the set of natural numbers out of the original set of natural numbers.<\/p>\n
\nthe set {0, 2, 4, 6, 8, …}, and the set {1, 3, 5, 7, 9, …}. The size of both of those sets is the ω – which is also the size of the original set you started with. <\/p>\n