The most interesting thing about the BT paradox, for me, is its interplay with the Axiom Of Choice. Most of the mathematics I did as part of my PhD would fall apart without AC, but I still remember being very unsettled by BT.
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 funny things is that the same result for an orbit on a sphere doesn't need the Axiom Of Choice at all. It just uses the concept of actual infinity.
"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.
The funny things is that the same result for an orbit on a sphere doesn't need the Axiom Of Choice at all. It just uses the concept of actual infinity.
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 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.
It is the theorem from a dead-tree math-textbook where it is used as a step in a Banach-Tarski Paradox proving.
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
Proof:
A = H(a)x; B = H(a')x; C = H(b)x; D = H(b')x
The theorem statement follows from the fact that the free group H can be decomposed into 4 parts:
H(a),H(a'),H(b),H(b')
and doubled by rotations:
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:
I can't follow either of the proofs yet, but one question does pose itself: given a decomposition of the sphere, don't you get a decomposition of the ball, just by projecting towards the origin?
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.
So would I. I think it's wrong; I'm pretty sure that the Hausdorff paradox, which is kinda-sorta the equivalent on the sphere, requires some Choice. But maybe I'm confused.
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.)
Infinite set paradoxing has become a morbid infection that is today spreading in a way that threatens the very life of probability theory, and requires immediate surgical removal. --- E.T. Jaynes
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 Banach-Tarski Gyroscope is an intricate mechanism believed to have been constructed using the Axiom of Choice. On each complete rotation counterclockwise, the Banach-Tarski Gyroscope doubles in volume while maintaining its shape and density; on rotating clockwise, the volume is halved. When first discovered, fortunately in the midst of interstellar space, the Banach-Tarski Gyroscope was tragically mistaken for an ordinary desk ornament. Subsequently it required a significant portion of the available energy of the contemporary galactic civilization to reverse the rotation before nearby star systems were endangered; fortunately, the Banach-Tarski Gyroscope still obeys lightspeed limitations on rotation rates, and cannot grow rapidly once expanding past planetary size. After the subsequent investigation, the Banach-Tarski Gyroscope was spun clockwise and left spinning."
Feynman's strength (well, one of them) was that he insisted on a simple example to follow during an experiment. This is an incredibly powerful technique for understanding complex calculations, and no doubt many hackers here use it when doing a code walk-through.
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."
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.
(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.)
Be careful, friends... digging too deep into set theory is one of the surest ways to go insane. You quickly reach the edge of what you consciousness can handle.
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,
