TITLE: Mathematical paradoxes?
QUESTION [6 upvotes]: What are some interesting mathematical paradoxes?
What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc..
Edit: As an additional restriction, let us focus on paradoxes that are not already in the list at:
https://secure.wikimedia.org/wikipedia/en/wiki/Category:Mathematics_paradoxes

REPLY [3 votes]: One of the consequences of Goedel's incompleteness theorem is that if $T$ is a finitely axiomatizable theory of arithmetic, then 

$T$ proves that $T$ is consistent

if and only if 

$T$ is inconsistent!

The reason is that an inconsistent theory proves anything, and a consistent theory never proves its own consistency.