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 ?)
Best Answer
The reason the Banach-Tarski paradox seems paradoxical is because of the following naive argument: surely the volume of a ball is the same as the sum of the volume of any possible decomposition of that ball into finitely many pieces, which is in turn not the same as the volume of two balls. More precisely, surely the total measure ought to remain invariant.
And the reason the Banach-Tarski paradox is a theorem is that the intermediate pieces it uses are very weird: in particular, they do not have volume. (More precisely, they are non-measurable.) So the naive argument breaks down completely, but naive arguments break down all the time in mathematics.
A more focused version of your question might be: how weird or pathological should I regard a non-measurable set as being? Well, of course they are weird, but they aren't weird to the point that they're a good reason (in my opinion) to reject the axiom of choice. One can construct non-measurable sets using the weaker ultrafilter lemma, which I happen to be extremely fond of, so I embrace them out of necessity.
Edit: You might also be interested in hearing Terence Tao's thoughts; he's written about Banach-Tarski several times and has enlightening things to say.