Banach-Tarski paradox

[DeadWolfe]

Is this paradox a valid refutation/disproof of the axiom of choice?

I don't know very much about it myself, but I thought it might make an interesting topic.

[Hurkyl]

Only if you have reason to think the result is wrong, not just surprising.

The axioms generally used by mathematicians do not prove the result wrong, if that's what you're asking.

[selfAdjoint]

The point is that the result flies in the face, not only of naive intuition, but of physics. Since it only uses beginner measure theory plus the axiom of choice, it seems tht the outrage is directly due to the AoC.

[matt grime]

Whether one chooses to accept the aciom of choice is largely personal preference. To many of us it is *obvious* that a vector space always has a basis. So we want it. It also leads to some weird stuff.

www.dpmmms.cam.ac.uk/~wtg10

then follow the links to his lecture to the philosophical society, where he gives a couple of examples where the axiom of choice ought to be true and one where it isn't. I believe Devlin has a thought experiment in one of his monthly articles which indicates some of the subtlety too.

EDIT:

Acutally the Devlin thing is on the axiom of constuctibilty and the continuum hypothesis (how many real numbers are there) but it's fairly close to some of this stuff, and reasonably illuminating to the layman.

[HallsofIvy]

"The point is that the result flies in the face, not only of naive intuition, but of physics. Since it only uses beginner measure theory plus the axiom of choice, it seems tht the outrage is directly due to the AoC."

"naive intuition" is just another name for "common experience" and one simply does not have common experience with the kind of sets used in the Banach-Tarski theorem. It does not "fly in the face" of physics since physics has nothing to do with this. The types of sets used are not in any sense "physical". I don't see any "outrage".

[matt grime]

http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html


here it is corrected and in full