Why is Banach–Tarski's paradox so interesting?
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.
The reason it is interesting is in the exact nature of the isomorphism involved in the Banach Tarski paradox, namely, isomorphism up to rigid motion of $\mathbb{R}^3$.
What the paradox says is that a solid ball in $\mathbb{R}^3$ can be partitioned into a finite number of sets $A_1,…,A_K$, and those sets $A_1,…,A_K$ can be moved around by rigid motions of $\mathbb{R}^3$, so that they consitute two solid balls each of the same size as the original. That is not what is going on with $[0,1]$ and $[0,2]$.
Added to address some comments: The Banach-Tarski paradox does not occur in $\mathbb{R}$ or in $\mathbb{R}^2$. The underlying reason for this difference with $\mathbb{R}^3$ is that the group of rigid motions of $\mathbb{R}$ or of $\mathbb{R}^2$ is amenable, whereas the group of rigid motions of $\mathbb{R}^3$ contains a free group of rank $>1$ and therefore is not amenable.
The map from $[0,1]$ to $[0,2]$ does not preserve measure. The map which "recomposes" the unit ball does. Which reinforces the point that "outlaw sets" can disobey many "laws" (in this case, outlaws means non-measurable, and disobey a law means to have a strange behavior under measure preserving maps).
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!).