Gödel's Incompleteness Theorems
Kurt Gödel's 1931 proof that any sufficiently expressive mathematical system contains true statements it cannot prove — math has a built-in limit.
Origin
Kurt Gödel (1906–1978) published the theorems at twenty-five. The first says: in any consistent formal system rich enough to do basic arithmetic, there exist true statements that the system can express but cannot prove. The second says: such a system cannot prove its own consistency. The result wrecked the early-20th-century dream (championed by Hilbert and Russell) of putting all of mathematics on a fully provable foundation. Gödel was a friend of Einstein at Princeton; he died of starvation in 1978 after refusing to eat food not prepared by his wife, who had been hospitalized.
Modern usage
Frequently invoked (often loosely) in arguments about the limits of AI, the limits of science, and free will. Douglas Hofstadter's Gödel, Escher, Bach (1979) made it the centerpiece of a pop-math classic. Strict mathematicians wince when 'Gödel' is used as a gesture rather than a result.
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