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.)
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.)
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.)
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.)