Paradox

A paradox is a statement that seems to lead to a logical self-contradiction, or to a situation that contradicts common intuition. The identification of a paradox based on seemingly simple and reasonable concepts has often led to significant advances in science, philosophy and mathematics.

See also:

Banach-Tarski paradox: Cut a ball into 5 pieces, re-assemble the pieces to get two balls.
Berry paradox: What is "The first number not nameable in under ten words"?
Burali-Forti paradox
Epimenides paradox: A Cretan says "All Cretans are liars".
EPR paradox: Can far away events influence each other in quantum mechanics?
Fermi paradox: If there are many other sentient species in the Universe, then where are they? Shouldn't their presence be obvious?
Grandfather paradox: You travel back in time and kill your grandfather before he meets your grandmother, resulting in you never being conceived.
Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.
Liar paradox: "This statement is false."
Newcomb's paradox: how do you play a game against an omniscient opponent?
Olbers' paradox: The sky should be bright because there's a star in every direction.
Omnipotence paradox: does omnipotence include the ability to make something even more powerful than yourself?
Raven paradox: Observing a red apple increases the likelihood of all ravens being black.
Russell's paradox: The barber who shaves all people who don't shave themselves.
Simpson's paradox: Data sets support opposite hypothesis when taken together than when taken separately.
Twin paradox: When the travelling twin returns, he's younger and older than his brother who stayed put.
Unexpected hanging paradox: The day of the hanging will be a surprise, so it can't happen at all.
Voting paradox: Non-transitive collective preferences.
Zeno's paradoxes: When you reach the turtle's spot, it has already advanced a bit, so you can never catch it.