He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal.
[2] Cohen was born in Long Branch, New Jersey in 1934, into a Jewish family that had immigrated to the United States from what is now Poland; he grew up in Brooklyn.
[5][6] In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at the University of Rochester for a year.
[10] On June 2, 1995, Cohen received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden.
In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory.
He was awarded the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper "On a conjecture by Littlewood and idempotent measures",[13] and lends his name to the Cohen–Hewitt factorization theorem.
Angus MacIntyre of the Queen Mary University of London stated about Cohen: "He was dauntingly clever, and one would have had to be naive or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s."
He went on to compare Cohen to Kurt Gödel, saying: "Nothing more dramatic than their work has happened in the history of the subject.
"[14] Gödel himself wrote a letter to Cohen in 1963, a draft of which stated, "Let me repeat that it is really a delight to read your proof of the ind[ependence] of the cont[inuum] hyp[othesis].
"[15] While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he had "had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory.
"[16] A point of view which the author [Cohen] feels may eventually come to be accepted is that CH is obviously false.
The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe.
Shortly before his death, Cohen gave a lecture describing his solution to the problem of the continuum hypothesis at the 2006 Gödel centennial conference in Vienna.