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paradox
A paradox is an assertion that is apparently selfcontradictory, though based on a valid deduction from acceptable premises.
Paradoxes typically lead to a reevaluation of the axioms of mathematics. Even after axioms are assumed so that the paradox is averted, the statement is still usually referred to as a paradox.
Occasionally, one may refer to a surprising result as a paradox. Such is the case in the birthday paradox, which is not apparently selfcontradictory.
Examples of paradoxes include:

BanachTarski paradox

BuraliForti paradox

Cantor’s paradox

Russell’s paradox

Simpson’s paradox

Zeno’s paradox
Related:
GalileosParadox
Synonym:
paradoxical, paradoxically, dilemma
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