A puzzle about barbers helped expose a crack in the foundations of mathematics
Does the barber who shaves exactly those men who do not shave themselves shave himself? Either answer contradicts itself. Riddles like this are more than party tricks: Russell's version about sets forced mathematicians to rebuild the foundations of logic and set theory.
A paradox starts from premises that look true and reasoning that looks valid, yet ends somewhere contradictory or unacceptable. Three ingredients recur: self-reference, contradiction and endless regress, often joined by circular definitions or muddled levels of abstraction. Self-reference alone is harmless; saying of a sentence that it is written in English is simply true, and claiming it is written in French is simply false. Trouble begins when a statement undermines itself, as in the liar: if this sentence is false, then it is true, which makes it false, and so on without end.
Some paradoxes rest on a hidden error or a hasty assumption. In the riddle of a surgeon who cannot operate on an injured boy because he is her son, the only puzzle is the listener's assumption that the doctor is a man. Others are veridical: results that sound wrong but are correct. The genuinely stubborn ones sit at the edges of language and logic and dissolve only when the framework is extended.
Russell's paradox asks whether the collection of all collections that do not contain themselves contains itself. It showed that building set theory by equating sets with properties was flawed, and it shaped modern logic. Curry's paradox is harder still, since it resists fixes to the underlying system. Even paradoxes that are invalid arguments sharpen critical thinking.
Beyond logic, the ship of Theseus asks whether a vessel rebuilt plank by plank is still the same ship. The grandfather paradox imagines a time traveller preventing his own birth; its apparent force comes from quietly treating the past he visits as both the same as and different from the one leading to his own present. M. C. Escher turned paradox into pictures, drawing walls that serve as floors from other angles and staircases that climb forever.
Source: Paradox
