Paul Cohen
1934–2007, Long Branch, New Jersey, United States, Palo Alto, California, United States
American mathematician who invented the method of forcing in 1963 to prove that the continuum hypothesis and the axiom of choice are independent of Zermelo–Fraenkel set theory, winning the 1966 Fields Medal.
Paul Cohen grew up in Brooklyn, graduated from Stuyvesant High School at sixteen, and entered graduate study at the University of Chicago without finishing a bachelor's degree. He earned his doctorate in 1958 under Antoni Zygmund on the uniqueness of trigonometric series and, before turning to logic, made a major breakthrough in harmonic analysis on Littlewood's conjecture for idempotent measures (receiving the 1964 Bôcher Memorial Prize). He joined Stanford University in 1961.
Although trained as an analyst rather than a logician, Cohen set out in late 1962 to resolve Hilbert's first problem — Cantor's continuum hypothesis. Kurt Gödel had shown in 1940 that neither the continuum hypothesis nor the axiom of choice can be disproved from Zermelo–Fraenkel set theory (ZF). In April 1963 Cohen invented the method of forcing, adjoining generic sets to a ground model of set theory to build models in which the continuum hypothesis fails (while ZFC holds) and models in which the axiom of choice fails (while ZF holds). Combined with Gödel's result, this proved both statements are independent of the standard axioms.
Cohen received the Fields Medal in 1966 — the only Fields Medal awarded for work in mathematical logic — and the National Medal of Science in 1967. Forcing became the central technique of modern set theory, and Cohen remained a professor at Stanford until his retirement in 2004.
Workplaces: Institute for Advanced Study, Princeton, Stanford University
United States