Implication and Interpretation of Banach Tarski
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 ?)
Solution 1:
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.
Solution 2:
It seems that the vast majority of mathematicians consider Banach-Tarski an acceptable price to pay for having the Axiom of Choice available. Its practical consequences are at least twofold, one general and one more specific:
Be wary of your intuition about sets that are so weird that you need the axiom of choice to construct them. The general consensus seems to be that the Banach-Tarski components behave non-intuitively, but not paradoxically.
In the presence of the Axiom of Choice one has to accept that there is no well-behaved measure on all subsets of $\mathbb R^n$ (where "well-behaved" means something like being invariant under isometries, at least finitely additive, and assigning a nonzero finite measure to an ball of finite radius). This entails a plethora of "assume such-and-such is measurable" premises in many theorems, but again this is generally treated as an acceptable price to pay for having choice.
Arguably, it even leads to a more general and useful theory that the main example of a measure is not defined on the full power set. Otherwise, the standard development of everything would just assume that all subsets are measurable, and then one would have trouble applying the theory to special measures that one wanted, for one reason or another, to be partial.
Solution 3:
Qiaochu's answer is great, and reflects exactly what a mathematician should feel about the Banach-Tarski theorem.
I would like to add another opinion, as someone who works mostly in a choiceless context I can assure you that mathematics has many surprises in store for you once you give up the choice needed for Banach-Tarski.
You might end up with the bizarre universe in which there are no free ultrafilters on $\mathbb N$; the real numbers might be a countable union of countable sets; or it might be possible to cut the real numbers into more non-empty parts than elements.
There is always a "paradox", which is really just a counter-intuitive theorem, hiding in the dark corners of the universe. It tells you, in the philosophical level, just one thing:
Our intuition is completely developed by history and the axioms we are used to work with. Once you are completely used to the axiom of choice there is no surprise in the Banach-Tarski theorem, much like there is no surprise in Gödel's incompleteness theorems or in Cantor's theorem about the uncountability of the real numbers.
These are all theorems that shook the foundations of mathematics and caused people to shake their heads in disbelief, but eventually these theorems were accepted and nowadays people don't fuss about Banach-Tarski because it's one of the first thing presented in a course about measure theory: You can't measure everything in a translation-invariant way and with countable completeness.
The main issue is that in any strong enough theory there will be unexpected results, which is why the Banach-Tarski theorem - while very surprising - should not deter you from the axiom of choice, which makes infinitary things easier to deal with.