Paul Cohen

Paul Cohen was an American mathematician who in 1963 resolved one of the most famous open problems in the foundations of mathematics, the status of the continuum hypothesis, a statement about the possible sizes of infinite sets first posed as a problem by Georg Cantor in the nineteenth century and placed first on David Hilbert's influential 1900 list of unsolved mathematical problems. Cohen developed an entirely new mathematical technique, called forcing, for constructing alternative mathematical universes in which a given statement can be made either true or false without producing any logical contradiction, and using this technique he proved that both the continuum hypothesis and the closely related axiom of choice are independent of the standard Zermelo-Fraenkel axioms of set theory, meaning that these standard axioms are simply not strong enough to prove or disprove either statement one way or the other. The result built on and complemented earlier work by the logician Kurt Godel, who had shown that neither statement could be disproven from the standard axioms, together completing the picture that both statements are genuinely independent of them. Cohen was awarded the Fields Medal in 1966 for this achievement, a prize that as of the time of this description remains the only Fields Medal ever given for work specifically in mathematical logic, and he also received the United States National Medal of Science in 1967 for his contributions to mathematics.