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.

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.

The simplest one is this: Suppose you have a sphere. You can take that sphere, and slice it into a *finite* number of pieces. Then you can take those pieces, and re-assemble them so that, without any gaps, you now have *two* spheres of the exact same size as the original.

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* sphere.

This sure as heck *seems* 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 that tries to explain it in terms of density. There's a very cool visualization of it here, 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* 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.

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.

Here's an equivalent thing, which is a bit simpler to think about:

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.

Still counterintuitive, but slightly easier.

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,

the 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.

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* 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.

The problem with Banach-Tarski is that we tend to think of it less in mathematical terms, and more in concrete terms. It's often described as something like "You can slice up an orange, and then re-assemble it into two identical oranges". Or "you can cut a baseball into pieces, and re-assemble it into a basketball." Those are both obviously ridiculous. But they're ridiculous because they violate one of our instinct that derives from the conservation of mass. You can't turn one apple into two apples: there's only a specific, finite amount of *stuff* in an apple, and you can't turn it into two apples that are identical to the original.

But math doesn't have to follow conservation of mass in that way. A sphere doesn't *have* a mass. It's just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.

Going further down that path: Banach-Tarski relies deeply of the axiom of choice. The "pieces" that you cut have *non-measurable* volume. You're "cutting" from the collection of points in the sphere in a way that requires you to make an *uncountably infinite* number of distinct "cuts" to produce each piece. It's effectively a geometric version of "take every other real number, and put them into separate sets". On that level, because you can't actually *do* anything like that, it's impossible and ridiculous. But you need to remember: we aren't talking about apples or baseballs. We're talking about sets. The "slices" in B-T aren't something you can cut with a knife - they're infinitely subdivided, not-contiguous pieces. Nothing in the real world has that property, and no real-world process has the ability to cut like that. But we're not talking about the real world; we're talking about abstractions. And on the level of abstractions, it's no stranger than creating two copies of the set of real numbers.