Banach–Tarski Paradox

en.wikipedia.org

11 points·noobermin··5 comments

This comic [1] has a pretty solid explanation of it underneath. And if you don't like the Axiom of Choice (or think that this paradox is a good reason to reject it), read the things that can happen when you don't have it at [2].

[1] http://www.irregularwebcomic.net/2339.html [2] http://mathoverflow.net/a/70435

Are the proofs in [2] relying in the excluded middle? Like, "because I don't have the axiom of choice, and you would need axiom of choice to prove it true, it must be false" or something? I guess I am just confused how less axioms can lead to more results.

Well,

Most of the examples are all the lines of "without the axiom of choice, one could assume X and not reach a contradiction" - you could assume you had an infinite set with no countably infinite subsets etc.

Which is a little different than the OP, which says something must exist. Within reason, more axioms increase the number of things that definitely are the case but fewer axioms increase the number of things that might be the case.

It's basically a demonstration that sets with the axiom of choice and no other restrictions are wild things.

Of course, if one restricts consideration to "measurable sets" and other objects of standard geometry, the paradox can't happen.

And discrete spaces, I believe. And there are compelling reasons that the real world is better modeled by a very fine discrete space.

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