# Why can't Banach–Tarski make me a rich bastard?

There is a well-known proof in mathematics (the Banach–Tarski paradox) which states that any solid sphere can be cut into five pieces in such a way that you can reassemble the pieces into two solid spheres, each the same volume as the original. Though this result sounds absurd, the proof and the axioms upon which it depends are pretty much accepted by most mathematicians today. (Please let’s not start another 0.9999…=1-type argument regarding this. Just either accept it or ignore this thread.)

Why is it that this proof works well in theory, yet I have so far observed no one capitalizing on it in the real world by producing exact replicas of pearls, peas, bowling balls, or spheres of solid gold? In fact, it seems to me that I could manufacture as much gold as I want by starting with a tiny gold marble and cutting it, reassembling it into two marbles, melting them down, recasting them into a single larger marble, and then repeating this until I’ve got a gold ball the size of my house.

IANAM, but I’m gonna guess that the issue is that mathematics allows you to posit an infinite number of points in any volume. In real life, solids are composed of finite numbers of atoms. So there’s no one-to-one mapping between B-T’s solids and actual physical objects.

Two piddling things stand between you and infinite riches:
[ol]
[li]Your gold ball is made of indivisible atoms, meaning that there’s a certain volume that you can’t get smaller than. The Banach-Tarski decomposition requires you to decompose the sphere into two non-measurable sets; “non-measurable”, here, meaning that they don’t have a well-defined volume. Obviously, a physical decomposition into jillions of units with a given volume will still have a well-defined volume.[/li][li]Even if you could get around atomic indivisibility somehow, the dissection given in the proof uses the axiom of choice, which says that given any collection of “bins”, you can pick one element from each bin. Unfortunately, there are infinitely many “bins” that you have to pick from in the Banach-Tarski dissection, meaning that it’ll take you an awful long time to draw up the blueprints for your get-rich-quick scheme.[/li][/ol]
Hope this helps.

Furthermore, the pieces that the sphere would have to be disassembled into, which are then moved and reassembled, are not continuous pieces of the sphere. Consider the following analogy: Suppose you had an absolutely straight stick of length 1 of an absolutely consistent width. Suppose it was not composed of atoms but of some infinitely decomposable material. Suppose you could break up such a stick into two pieces by a single cut if you wanted to (and this cut could be any point on the stick). Suppose you were told to break it up into two pieces, one of which consisted of all the points on the stick which were exactly a rational length from one end of the stick and the other of which consisted of all the points on the stick that are exactly an irrational length from that same end of the stick. It would take you forever to do this, assuming it takes a finite time for one cut of the stick. Now, this isn’t quite what you have to do if you were to use the Banach-Tarski theorem to slice up a sphere. What you have to do for the Banach-Tarski theorem is harder than this.

The second part of your post has been pretty much answered. I just want to nitpick this bit. There are still debates going on over this, but here’s a rough idea of the functionally predominant[1] view:

A collection of axioms describe a certain kind of structure. Sometimes the structure is uniquely defined (“up to isomorphism”) like the natural numbers, and sometimes it’s more general like the notion of a group. This much is pretty uncontroversial.

For a long time now, sets have been used as the building blocks of everything else, but they’ve got some big problems. For one, to use a set-theoretical background you have to choose whether or not to use various axioms, and the oddest one is the Axiom of Choice, and its equivalents Zorn’s Lemma and the Well-Ordering Principle[2]. The problem arises that when you’re doing algebra it’s really nice to have Zorn’s Lemma around. In fact, it’s almost indispensible. However, when you’re doing analysis (think “calculus, but more so”) or point-set topology, the Axiom of Choice gives you weird things like non-measurable sets (as MikeS mentioned) and outright absurdities like Banach-Tarski. Even worse, algebraic topology gets a mixed bag. The upshot is that the Axiom of Choice is so nice for algebra that mathematicians generally accept the weirdness, but there are a few holdouts.

Anyhow, what if we never had to make that choice? Enter topoi. A topos is a kind of mathematical structure which does all sorts of nifty things. In particular, various topoi are really good at providing models for various kinds of set theory. Originally, of course, topoi arose from a field rooted in sets (usually thought of with Choice), and there’s a topos of sets, which is sort of like emulating a computer in its own software. You can also construct topoi of other variations on the notion of sets. These can, for instance, provide models of set theories which reject Choice.

So there’s a push on (among people who even think about such things anymore) to try to replant the whole tree of mathematics in topoi from the beginning. Now the question can be answered, “The Banach-Tarski paradox involves cutting infinitely finely, would take forever to make the choices the Axiom of Choice tells us can be made, and only holds in Choice topoi to begin with”. Of course, there’s no evidence on either side that Choice holds in whatever topos the real world occupies or not, but the Intuitionists (people who hold Banach-Tarski up as an absurdity, among other things) would suggest that the real world’s logic doesn’t contain Choice.

[1] Oddly enough, most modern mathematicians don’t pay much attention to mathematical philosophy. This is the view that (as far as I can tell) best describes “how mathematicians really do math”.

[2] The Axiom of Choice says that if you have any collection of sets indexed by another set, you can choose one element of each set. It’s nontrivial when the indexing set is very infinite.

Well-Ordering says that the elements of any set can be put into a line so that any subset contains a least element. Just try this with the real numbers to see how things start to go weird.

Zorn’s Lemma is harder to describe. Basically if you’ve got a notion of “greater than”, but not every pair of elements are comparable (think about (x,y) > (z,w) if x>y and z>w. Then (1,2) and (2,1) are incomparable), and for any sequence of increasing elements (each “greater than” the last) you can find an element that’s greater than all of the elements of the sequence, then the set contains at least one element that no element is greater than (though other elements may be incomparable).

Although all three are logically equivalent, mathematicians (well, those who are like this mathematician) are fond of noting that “The Axiom of Choice is obviously true, the Well-Ordering Principle is plainly false, and who knows about Zorn’s Lemma?”

I assume you know what’s yellow and equivalent to the axiom of choice.

Of course. I also know what’s purple and commutes, what’s green and very far away, and what’s grey and ignored by Grothendieck.

I take it that the “not every pair of elements are comparable” part is necessary, not just tolerable. If, for instance, I take the usual order relation on the set of real integers (all of which are comparable), I can find no such element which is not exceeded. However, if I take the set of real integers together with i, the imaginary number (which is not comparable with any real number), then I can say that i is my number which is not exceeded. Does it in general occur that the number which is not exceeded must be incomparable with some other number?

Also, is there any known well-ordering of the real numbers? Such a well-ordering would not necessarily imply the truth of the Well-Ordering Principle, which requires such of every set, but it would be interesting.

What if the indexing set is only slightly infinite?

and every sequence of increasing elements is bounded above. The problem with the real numbers is not partial vs. total order, but that there are unbounded increasing sequences.

You mean some well-ordering I can write down a deterministic formula for? I don’t know, and I doubt it. Well-orderings are very nice for some set theoretic constructions, but they tend to break every other property than ordering.

The Axiom of Finite Choice is a theorem in Zermelo-Fraenkel set theory (that is, it can be proven from the other axioms).

The Axiom of Countable Choice – when the indexing set is countably infinite – is axiomatic in ZF, but isn’t quite so problematic for certain applications, and it’s very difficult to come up with models of ZF which violate it.

My point was more to nudge the lay reader who may intuit from the case of a finite indexing set that the difficulties really lie in the transfinite indexing sets.

I would think that you can’t get something from nothing.

Count me in with the intuitionists on this one. You can’t divide one sphere into bits and put them back to form two spheres identical to the one you started with. So if you have some ‘fancy’ mathematical reasoning that says you can, then I say it’s flawed and I wish you a very nice time working out where that flaw lies.

If you still wanna claim you’re right, and that I’m an ignorant dolt who doesn’t understand the math, that’s okay. I still say it can’t be done. Want to prove me wrong? Fine. Do it. One demonstration that this can be done, and I’ll concede defeat. Otherwise, I won’t. And why the heck should I?

What next? A math theorem that proves you can lick your own elbows?

Intuition is often wrong, as is famously demonstrated here every once in a while when some intuitionist tries to argue his (incorrect) beliefs about 0.999…=1 or the Monty Hall problem.

You don’t need a math theorem for that; just a friend with a chainsaw.

The Banach-Tarski paradox does not say that it’s physically possible to cut a sphere into pieces and reassemble them into two spheres of the same size. it says that there exists a way of cutting a sphere into pieces which can be assembled into two spheres of the same size without having to change the shape of those pieces. This way of cutting up the sphere, even if you assumed an infinitely divisible matter, couldn’t be done in a finite time if you assume that cutting up matter takes some time for each cut. Furthermore, this method of cutting up the sphere is not specified by the theorem. It merely says that there exists such a method, somewhere out there among all the possible ways of cutting up spheres.

I’d settle for an approximation using a finite number of cuts.

Agreed. It’s sometimes right and it’s sometimes wrong. The point is whether it’s right or wrong in this specific instance. I say it’s right, and that this sphere dissection thing can’t be done. Would be perfectly happy for a demonstration to be arranged to show me I’m wrong.

ianzin writes:

> Would be perfectly happy for a demonstration to be arranged to show me I’m
> wrong.

Once again, the theorem does not claim that it’s possible to do a demostration. You keep asking for something that nobody claims is possible.

That pretty much sums up what little I know about set theory: that it consists of the parts that are so trivially obvious as to be pointless, the parts that are almost certainly false, and the rest which is incomprehensible.

That only works for certain values of ‘friend’.