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:

Allmigthy God paradox: The Allmigthy God is not able to make a very big and heavy rock, so big and so very heavy, that even He could not raise.
Arrow's paradox: You can't have all the attributes of an ideal voting system at once
Banach-Tarski paradox: Cut a ball into 5 pieces, re-assemble the pieces to get two balls, both of equal size to the first.
Barber paradox: The barber who shaves all men who don't shave themselves, and no-one else.
Berry paradox: What is "The first number not nameable in under ten words"?
Birthday paradox: What is the chance that two people in a room have the same birthday?
Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number which is smaller than itself.
Control paradox Man can never be free of control, for to be free of control is to be controlled by oneself.
Curry's paradox: "If this statement is true, then Y is true."
Epimenides paradox: A Cretan says "All Cretans are liars".
Einstein-Podolsky-Rosen 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.
the GZK paradox: high-energy cosmic rays have been observed which seem to violate the Greisen-Zatsepin-Kuzmin limit which is a consequence of special relativity
Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.
Horse paradox: All horses have the same color.
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: Can an omnipotent being create a rock too heavy to lift? Can an irresistible force move an unmovable object?
Predestination paradox: A man travels back in time and impregnates his great-great-grandmother. The result is a line of offspring and descendants, including the man's parent(s) and the man himself. Therefore, unless he makes the time-travel trip at all, he will never exist.
Raven paradox: Observing a red apple increases the likelihood of all ravens being black.
Richard paradox: A complete list of definitions of real numbers doesn't exist.
Russell's paradox: Is there a set of all those sets that do not contain 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.