Talk:804: Pumpkin Carving
I'm bringing over a comment from the blog, because it helps in understanding the Banach-Tarski theorem and Axiom of Choice.
- I realize that you probably aren’t checking these comments anymore, but I’m math grad student currently wasting time I shouldn’t be, and can’t resist answering this question.
- So the short answer is pretty much exactly what Jonathan said–the Axiom of Choice is in someways the “extra” axiom of ZFC set theory (in fact ZFC stands for “Zermelo–Fraenkel set theory *with the axiom of choice*”) and for a long period of time mathematicians were attempting to either
- (1) prove the axiom of choice as a consequence of the other axioms of set theory
- (2) discover a compelling reason why we should not accept the axiom of choice.
- The Banach-Tarski theorem was published at the height of the debate/research, and is still frequently the first thing sited by someone who doesn’t accept the axiom of choice (although most working mathematicians I know do accept the axiom of choice, in part because it just seems silly to handicap yourself unnecessarily).
- What makes the Banach-Tarski theorem seem so paradoxical is simply the fact that they show it is possible to cut a ball into a finite number of pieces (5, to be specific) and reassemble these pieces only using rotations and translations (ie, only by movements you can make with your own hands) to produce two balls, each identical in volume to the first–ie, in someways "1 [ball] = 2 [balls]", which certainly feels a bit shady.
- All of the other theorems and axioms on which the Banach-Tarski theorem relies are standard and relatively accepted areas of mathematics, while the axiom of choice (which is used at a key point of the proof to “choose” elements not fixed in place by particular nice rotations of the sphere) was not, so many people consider this a reason not to accept the axiom of choice.
- If we don’t accept the axiom of choice, it becomes impossible to pick those points, and the whole proof breaks down. In fact, it can also be shown that if we don’t accept the axiom of choice, there don’t exist any “unmeasurable” sets–sets such as those created in the cutting of the ball whose volume we can’t really talk about, as it would need to be add up to 1 (after all those 5 sets together form a ball of volume 1!) and add up to 2 (after all, those 5 sets together form two balls of volume 2!), which makes some people very happy with the idea of rejecting the axiom of choice.
- For myself, I actually *don’t* find the theorem very paradoxical at all–these 5 “pieces” into which the ball is decomposed are incredibly crazy, and nothing you could ever cut with a scalpel, or even a laser, no matter how good you are. Additionally, the theorem really follows relatively easily from a theorem everyone accepts, which is that you can similarly split up the group of rotations to four disjoint pieces and “reassemble” them via rotations into two copies of the group of rotations. It seems a little odd at first, but the thing to keep in mind that any time infinite things get involved, things are going to get a little odd (after all–what’s infinity + infinity? What about infinity/2?). Honestly, at this point, the thing which I find the most “paradoxical” about the Banach-Tarski paradox is that it can’t be done by dividing the ball into only 4 pieces, but this could be a sign I’ve been drinking the math-koolaid for too long. — Ashley