Kindly comments on The Hostile Arguer - LessWrong

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Comment author: CronoDAS 10 December 2014 06:46:25AM 3 points [-]

"Well, I know it sounds absurd that you'd be able to take a single object, disassemble it, and reassemble it to form two of the same object, but in fact it has been proven to be possible given infinite divisibility and something called the Axiom of Choice. If you're not familiar with that, I'd suggest reading a bit about set theory."

I actually can give you an "intuitive" justification of the Banach-Tarski theorem.

Suppose you have a rigid ball full of air. If you take half the air out and put it into another, identical ball, you now have twice the volume of air, at half the density. However, the points in a mathematical ball are infinitely dense - half of infinity is still infinity, so it turns out that if you do it just right, you can take out "half" of the points from a mathematical ball, put it inside another one, and end up with two balls that are both "completely full" and identical to the original one.

Comment author: Kindly 10 December 2014 05:52:01PM 3 points [-]

Your explanation suggests the wrong intuition for Banach-Tarski.

It's relatively easy to show that there's a bijection between the points contained in one ball and the points contained in two balls. (Similarly, there is a bijection between the interval [0,1] and the interval [0,2].)

The Banach-Tarski theorem proves a harder statement: you can take a unit ball, partition it into finitely many pieces (I think it can be done with five), and then rearrange those pieces, using only translations and rotations, into two unit balls.

(If there's a canonical weird thing about the theorem, it's that we can do this in three dimensions but not in two.)

Comment author: CronoDAS 11 December 2014 10:29:04PM 1 point [-]

Agreed; it's not a real justification, it's just something that makes it sound less absurd. (When you look at the theorem a little bit closer, the weird part becomes not that you can make two balls out of one ball, but that you can do it with just translations and rotations. And if you look really, really hard, the weird part becomes that you can't do it with only four pieces.)

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