TITLE: Why is Banach–Tarski's paradox so interesting?
QUESTION [35 upvotes]: Here is how I understand the Banach–Tarski paradox, based on the Wikipedia article : with a clever partitioning, one can decompose a solid ball into two solid balls, each identical to the first one.
I hear this paradox cited here and there a lot, but I don't really get what makes it so interesting.
For instance, it is easily accepted that $[0,1]$ and $[0,2]$ are isomorphic.
We can by the exact same reasoning show that a cube on $\mathbb{R}^3$ is isomorphic to twice itself.
To me, it seems that the 'Banach-Tarski paradox' immediately follows.
Have I missed something ?
PS : I am talking about the raw version of the paradox, not the numerous extensions that have been made of it, like showing that the decomposition can be chosen in such a way that the parts can be moved continuously into place without running into one another, etc.

REPLY [4 votes]: Banach-Tarkis paradox shows that it is impossible to have a Banach measure in 3 dimensions. This is a bit disappointing, because it would allow us to "measure" all sets, in some sense, if it existed. Also, this is surprising, because Banach measures exist in dimension $1$ and $2$.
A Banach measure is a much like a "normal" measure, except:

It is only required to be finitely additive.
It is supposed to be defined for all sets and be preserved under isometries.

Such measures are less useful for the purpose of developing integration theory, and so on, but it is not clear a priori (at least to me) that they are less interesting than the "normal" measures.
As for the "isomorphism" between $[0,1]$ and $[0,2]$, this shows that we can't have a measure that is invariant under scaling, which is not surprising at all (it would be rather surprising if you had such invariance!).