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.