TITLE: Implication and Interpretation of Banach Tarski
QUESTION [20 upvotes]: As I understand, the Banach-Tarski paradox says a ball in 3-space
may be decomposed into finitely many pieces and reassembled into two balls each of
the same size as the original. Despite being called a paradox it is of course a theorem. 
Looking at the proof, it seems to rely heavily on the Axiom of Choice. However since the consequences of not accepting the Axiom of Choice seem even more weird, I am wondering whether the more experienced Mathematicians here find the implication of Banach-Tarski a perfectly acceptable Theorem, or whether it shows that ZF with Choice is actually ultimately pathological ? ( i.e. does it just seem a weird from a perspective that is not mathematically mature enough ?)

REPLY [3 votes]: As something of an aside: it's important to remember that Banach--Tarski is not merely some artifact of choice.  Assuming choice, the BT paradox holds for some group actions (e.g. $SO(3)$ acting on $\mathbb R^3$), but not for others (e.g. $SO(2)$ acting on $\mathbb R^2$).  Indeed, there is an intrinsic group-theoretic property that governs whether or not the BT paradox can hold, namely amenability.  
The study of group actions and their interaction with measure theory is an entire branch of mathematics; see 
e.g. Topics in orbit equivalence, by Kechris and Miller.   There is much more to this topic than just the use of choice.