

Banach–Tarski Paradox - noobermin
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

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tgb
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](http://www.irregularwebcomic.net/2339.html)
[2] [http://mathoverflow.net/a/70435](http://mathoverflow.net/a/70435)

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kazagistar
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.

~~~
joe_the_user
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.

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joe_the_user
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.

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

