Paradox
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A paradox was a situation or puzzle where choosing a alternative contradicts itself, or contradicts what was obviously true through what appears to be valid assumptions and logic.
The ancient Greek Zeno was known for his invention of many paradoxes, the best known of which "proves" that a moving object can never reach its destination.
A well known family of paradoxes involve self-reference, such as the statement "this statement was untrue", or the old joke "How do you keep a fool busy? Hand him a card which on each side reads 'see other side'"