Paul J. Cohen | |
|---|---|
| Born | (1934-04-02)April 2, 1934 Long Branch, New Jersey, U.S. |
| Died | March 23, 2007(2007-03-23) (aged 72) Stanford, California, U.S. |
| Alma mater | University of Chicago (MS,PhD) |
| Known for | Cohen forcing Continuum hypothesis |
| Awards | Bôcher Prize (1964) Fields Medal (1966) National Medal of Science (1967) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Stanford University |
| Thesis | Topics in the Theory of Uniqueness of Trigonometrical Series (1958) |
| Doctoral advisor | Antoni Zygmund |
| Doctoral students | Peter Sarnak |
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that thecontinuum hypothesis and theaxiom of choice areindependent fromZermelo–Fraenkel set theory, for which he was awarded aFields Medal.[1][2]
Cohen was born inLong Branch, New Jersey in 1934, into aJewish family that had immigrated to the United States from what is nowPoland; he grew up inBrooklyn.[3][4] He graduated in 1950, at age 16, fromStuyvesant High School in New York City.[1][4]
Cohen next studied at theBrooklyn College from 1950 to 1953, but he left without earning hisbachelor's degree when he learned that he could start his graduate studies at theUniversity of Chicago with just two years of college. AtChicago, Cohen completed his master's degree in mathematics in 1954 and hisDoctor of Philosophy degree in 1958, under supervision ofAntoni Zygmund. The title of his doctoral thesis wasTopics in the Theory of Uniqueness of Trigonometrical Series.[5][6]
In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at theUniversity of Rochester for a year. He then spent the academic year 1958–59 at theMassachusetts Institute of Technology before spending 1959–61 as a fellow at theInstitute for Advanced Study at Princeton. These were years in which Cohen made a number of significant mathematical breakthroughs. InFactorization in group algebras (1959) he showed that any integrable function on a locally compact group is the convolution of two such functions, solving a problem posed byWalter Rudin. InCohen (1960) he made a significant breakthrough in solving the Littlewood conjecture.[7]
Cohen was a member of theAmerican Academy of Arts and Sciences,[8] the United StatesNational Academy of Sciences,[9] and theAmerican Philosophical Society.[10] On June 2, 1995, Cohen received anhonorary doctorate from the Faculty of Science and Technology atUppsala University,Sweden.[11]
Cohen is noted for developing a mathematical technique calledforcing, which he used to prove that neither thecontinuum hypothesis (CH) nor theaxiom of choice can be proved from the standardZermelo–Fraenkel axioms (ZF) ofset theory. In conjunction with the earlier work ofGödel, this showed that both of these statements arelogically 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 the most widely known 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 theFields Medal in mathematics in 1966, and also theNational Medal of Science in 1967.[12] The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic, as of 2022.
Apart from his work in set theory, Cohen also made many valuable contributions to analysis. He was awarded theBôcher Memorial Prize inmathematical analysis in 1964 for his paper "On a conjecture byLittlewood andidempotent measures",[13] and lends his name to theCohen–Hewitt factorization theorem.
Cohen was a full professor of mathematics atStanford University. He was an Invited Speaker at theICM in 1962 in Stockholm and in 1966 in Moscow.
Angus MacIntyre of theQueen 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 toKurt 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]. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play."[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. Indeed, they thought you had to be slightly crazy even to think about the problem."[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 theaxiom 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. Similarly with the higher axioms of infinity. Now is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely thepower set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from thereplacement axiom can ever reach.
Thus is greater than, where, etc. This point of view regards as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.
An "enduring and powerful product" of Cohen's work on the continuum hypothesis, and one that has been used by "countless mathematicians"[16] is known as"forcing", and it is used to construct mathematical models to test a given hypothesis for truth or falsehood.
Shortly before his death, Cohen gave a lecture describing his solution to the problem of the continuum hypothesis at the 2006 Gödel centennial conference inVienna.[17]
Cohen and his wife, Christina (née Karls), had three sons. Cohen died on March 23, 2007, inStanford, California, after suffering fromlung disease.[18]
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