The Banach–Tarski paradox
en.wikipedia.org
en.wikipedia.org
The existence of nonmeasurable sets, such as those in the Banach–Tarski paradox, has been used as an argument against the axiom of choice. Nevertheless, most mathematicians are willing to tolerate the existence of nonmeasurable sets, given that the axiom of choice has many other mathematically useful consequences.
"the axiom of choice has many other mathematically useful consequences." is an understatement. It's like a life in Space. You can live in Space (in a specially design isolated constructions e.g., a space station); you even can go outside (not for long and only if you are wearing a space-suit) but It is much much easier to live on the Earth.
I had to read this several times before I understood what you were saying. But you are right. The point is that you can get what wikipedia calls a paradoxical decomposition of a groups of rotations, and that doesn't require choice. Converting that into a proof of B-T requires taking a member of each equivalence class, and that does require choice.
The proof on WikiPedia is pretty good, actually:
http://en.wikipedia.org/wiki/Banach-Tarski_paradox#A_sketch_...
The point about the orbit on a sphere not needing AC is really step 1 of the proof there, except using rotations on a sphere as the "location" of the free group.
As a side issue, B-T is proof that there isn't a finitely additive, isometry-invariant measure on R^3 that's defined for all subsets. Interestingly, there is such a measure on R^2, although it's now 26 years since I proved it.
On the other hand, you don't need any Choice for the Sierpinski-Mazuriewicz paradox, which says that there's a subset A of the plane, which can be written as the disjoint union of two "smaller" subsets B and C, but where A, B, and C are all congruent. (But A is countable, so this isn't nearly as startling as Banach-Tarski.) Specifically, let T be translation by (1,0), and let R be rotation about the origin by 1 radian; then let A be the set of points you can get from (0,0) by applying some sequence of T and R, let B consist of the points where the last operation in the sequence was T rather than R, and let C be everything else. Then B = T(A) and C = R(A). (The key point here is that it turns out that the sequence of Ts and Rs is unique, so that B is well-defined.)
The textbook is not in English. Here's a translation of the theorem:
An orbit O [1] can be decomposed into 4 sets: A, B, C, D. Using rotation these sets can be combined into 2 orbits:
A ∪ aB = O; C ∪ bD = O
A = H(a)x; B = H(a')x; C = H(b)x; D = H(b')x
H(a),H(a'),H(b),H(b') H = aH(a') ∪ H(a); H = bH(b') ∪ H(b)__ [1]: The term `orbit` is used in the same sense as in [2]. `H` is a free group similar to the one from the step 3 in [2], and `a`, `b` are rotations defined similar to Step 2 in [2] i.e., they are generators of H:
H = {e}∪H(a)∪H(a')∪H(b)∪H(b')
aa' = e; bb' = e
Of course what happens at the origin itself is not clear. Perhaps there's no way of getting around that difficulty. If you could deal with that, however, a proof on the sphere would be equivalent to a proof on the ball.
The result for the sphere are easily generalized on a ball (first without a center, then with the center).
It applies to other theories as well. Giving real-life meaning to a construction based on infinity and then shouting "paradox!" seems counter-productive.
The problem was that the mathematicians fell into the trap Feynman laid and gave overly simple descriptions. Feynman seduced them into trying to give physical descriptions when it was, in truth, important to think about the details.
Banach-Tarski is a specific example of that, as many of you will know. I think Feynman knew full well what he was doing - just one of the games he played.
The technique has its limitations too. Restricting yourself to consider only the physical loses the power of abstraction. Many of you know that too, although perhaps you don't think of it that way. When you extract a method you're using abstraction - the method may not represent something physical. Feynman's technique can limit you there.
The story comes from "Surely You're Joking, Mr Feynam", in the chapter called "A Different Box of Tools."
http://www.multitran.ru/c/m.exe?a=DisplayParaSent&fname=...
Feynman makes a big deal about having unusual tools in the box - they let you solve problems that others find intractable. That's one real lesson. Acquire a wide variety tools, and know how to use them. Functional, Object Oriented, Imperative, Logic, Database, etc. They all have their place, and knowing them, really knowing them, gives you enormous flexibility.
Never stop learning new tricks.
(Speaking of Stan Wagon and beautiful mathematical exposition, his drably-titled paper "Fourteen proofs of a result about tiling a rectangle" is lovely and quite accessible. You can find it on the web with a bit of googling; I don't know whether any copy you can find in this way is legal.)
Attempts were once made to show that Euclid's fifth postulate could be deduced from the others by assuming it was false, and seeing where that led. A perfectly valid technique, as you know. Several assumed that it succeeded, because the conclusions were so obviously ludicrous, and from that it was assumed that Euclid's geometry was the only one.
Of course, now we know that the conclusions weren't contradictions, and are "simply" non-Euclidean geometries.
"Counter-Intuitive" is not the same as "Contradiction."
Just because the Axiom of Choice leads to such counter-intuitive results it doesn't mean they're not true. Just because infinities lead to bigger infinities (via Cantor), flasks of finite volume but infinite surface area,
http://community.tes.co.uk/forums/t/277351.aspx
and hotels that never run out of room,
http://mathforum.org/kb/message.jspa?messageID=5584228&t...
it doesn't mean that they aren't useful, or don't, in some sense, exist.