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Paul Joseph Cohen (April 2, 1934 – March 23, 2007)^{[1]}^{[2]} was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.
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Early years
Cohen was born in Long Branch, New Jersey, into a Jewish family. He graduated in 1950 from Stuyvesant High School in New York City.^{[2]}
Cohen next studied at the Brooklyn College from 1950 to 1953, but he left before earning his bachelor's degree when he learned that he could start his graduate studies at the University of Chicago with just two years of college. At Chicago, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy degree in 1958, under supervision of the Professor of Mathematics, Antoni Zygmund. The subject of his doctoral thesis was Topics in the Theory of Uniqueness of Trigonometric Series.^{[3]}
His contributions to mathematics
Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH), nor the axiom of choice, can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms. In this sense, the continuum hypothesis is undecidable, and it is probably the most widelyknown example of a natural statement that is independent from the standard ZF axioms of set theory.
For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967^{[4]}. The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic.
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