This statement is false. True or false?
Does the set of all sets that don't contain themselves contain itself?
Does the set of all sets that don't contain themselves contain itself?
[1] http://en.wikipedia.org/wiki/Continuum_hypothesis
[2] http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
While it's a very interesting thing to think about, there is one sense in which the paradox is trivially resolvable; while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe, and you certainly can not construct two real spherical shells made of atoms of the same size as one original spherical shell in the real world. (Whether the universe is continuous is an open problem, but we know matter isn't.) A great deal of the reason why the mind rebels at the Banach-Tarski paradox is that it is built on a thoroughly aphysical axiom; that the aphysical axiom permits aphysical (and therefore counterintuitive) results is not that surprising.
By no means is that a criticism of BT, it is simply its nature. My personal position on constructivist vs platonic vs. blah blah blah is that in math, as long as you specify which axiom set you are using up front, there is nothing to be emotional about. (Except that I will reserve a special place of interest for whatever mathematics it turns out to be that precisely represents the real world. Alas, this is still a work in progress, though we can point to at least some characteristics of it.)
- The exact same is true if you take the continuum hypothesis as an axiom. Both the axiom of choice and the continuum hypothesis are independent of the other standard ZF axioms
"If you do not take the axiom of choice, it is simply false"
- That is incorrect. It is only independent. To say it is false, another axiom(s) would need to contradict it.
"while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe"
- neither are the real numbers. would you like to claim that pi, or the square root of 2, for that matter, also don't exist? How about the number 1?
Therefore the truth of Banach-Tarski has nothing to do with any statement you can readily make about the integers, which would include any statement that you can make about anything that can be done on a Turing machine.
Incidentally the whole debate about constructivism is much more subtle than you likely appreciate. For instance it is trivial to prove that almost all real numbers that exist, cannot be written down. (Proof, I can easily write out a countably infinite set of statements that could define real numbers. So the set of real numbers that can be written down is countable. But there are an uncountable number of real numbers, so almost all cannot be written down.) In what sense do said real numbers actually exist?
If our axioms say that numbers must exist, which have in a very real sense no actual existence, then are those axioms capturing what we want to mean by "exist"?
If you say it does not, then you're a constructivist who hasn't thought carefully enough yet. If you say it does make sense, then you've been brainwashed by decades of standard mathematical presentations into accepting a definition of "existence" that makes no sense to the average lay person. :-P
Nitpick: If you do not choose the axiom of choice, then Banach-Tarski is independent, not false. So you can't prove that it's true and you can't prove that it's false.
None of this involves the kind of actual (or falsidical) paradoxes the incompleteness theorems do. (Note that the Banach-Tarski Paradox is a viridical paradox, as it is a true result that is only thought paradoxical because it contradicts naïve intuition. A falsidical paradox is one that shows a true flaw or limit in a logical system, such as Russell's Paradox is to naïve set theory. This terminology is due to Quine.)
that's a good distinction you make in the second paragraph. to me though, the viridical paradoxes are more interesting :-)
It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate.
>also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions.
I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.
I took two classes by George Boolos at MIT. They were a lot of fun, and Prof. Boolos was a rather strange fellow.
The Department of Linguistics and Philosophy has a memorial display case filled with some of Boolos's favorite puzzles. You should check it out if you're in the area.