Paradox

A paradox is a statement or set of statements that leads to a self-contradiction or a situation that defies intuition, even when premises appear reasonable. Paradoxes reveal ambiguities in language, hidden assumptions, and the limits of formal theories. They occur across philosophy, logic, mathematics, science, and everyday discourse, often prompting reassessment of definitions, methods, or axioms.

Etymology: The term comes from Greek paradoxos, meaning "contrary to expectation," from para- "beyond" and doxa "opinion."

Paradoxes are commonly grouped as logical, semantic, mathematical, or physical. Notable examples include the Liar paradox ("This statement is false"), which produces a contradiction about truth; Zeno's paradoxes (such as Achilles and the tortoise), which challenge notions of motion and infinity; Russell's paradox, which arises from considering the set of all sets that do not contain themselves and undermines naive set theory; the Banach–Tarski paradox, which in set theory allows a ball to be decomposed and reassembled into two congruent balls; and the grandfather paradox in discussions of time travel, where altering the past could negate the traveler's own existence.

Paradoxes serve as tools for clarifying concepts and testing theories. They often provoke refining definitions, adopting more precise logical frameworks, or changing underlying assumptions. Some paradoxes are resolved by language analysis, others by new axioms or theories, and some point to genuine limits of knowledge rather than to errors in reasoning.

Related topics include logic, philosophy, mathematics, and the study of language.