Paul Cohen

For other people named Paul Cohen, seePaul Cohen (disambiguation).Not to be confused withPaul Cohn.

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
Doctoral advisor	Antoni Zygmund
Doctoral students	Peter Sarnak

Paul Joseph Cohen(April 2, 1934 – March 23, 2007)was an American mathematician. He is best known for his proofs that thecontinuum hypothesisand theaxiom of choiceareindependentfromZermelo–Fraenkel set theory,for which he was awarded aFields Medal.

Early life and education

Cohen was born inLong Branch, New Jersey,into aJewishfamily that had immigrated to the United States from what is nowPoland;he grew up inBrooklyn.He graduated in 1950, at age 16, fromStuyvesant High SchoolinNew York City.

Cohen next studied at theBrooklyn Collegefrom 1950 to 1953, but he left without earning hisbachelor's degreewhen he learned that he could start his graduate studies at theUniversity of Chicagowith just two years of college. AtChicago,Cohen completed his master's degree in mathematics in 1954 and hisDoctor of Philosophydegree in 1958, under supervision ofAntoni Zygmund.The title of his doctoral thesis wasTopics in the Theory of Uniqueness of Trigonometrical Series.

In 1957, before the award of his doctorate, Cohen was appointed as an Instructor in Mathematics at theUniversity of Rochesterfor a year. He then spent the academic year 1958–59 at theMassachusetts Institute of Technologybefore spending 1959–61 as a fellow at theInstitute for Advanced Studyat 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.

Cohen was a member of theAmerican Academy of Arts and Sciences,the United StatesNational Academy of Sciences,and theAmerican Philosophical Society.On June 2, 1995, Cohen received anhonorary doctoratefrom the Faculty of Science and Technology atUppsala University,Sweden.

Career

Cohen is noted for developing a mathematical technique calledforcing,which he used to prove that neither thecontinuum hypothesis(CH) nor theaxiom of choicecan 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 independentof 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 Medalin mathematics in 1966, and also theNational Medal of Sciencein 1967.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 Prizeinmathematical analysisin 1964 for his paper "On a conjecture byLittlewoodandidempotent measures",and lends his name to theCohen–Hewitt factorization theorem.

Cohen was a full professor of mathematics atStanford University.He was an Invited Speaker at theICMin 1962 in Stockholm and in 1966 in Moscow.

Angus MacIntyreof theQueen Mary University of Londonstated 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."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."

Continuum hypothesis

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."

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 infinityis 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$\aleph _{1}$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$C$[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 axiomcan ever reach$C$.

Thus$C$is greater than$\aleph _{n},\aleph _{\ Omega },\aleph _{a}$,where$a=\aleph _{\ Omega }$,etc. This point of view regards$C$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.

— Cohen (2008)

An "enduring and powerful product" of Cohen's work on the continuum hypothesis, and one that has been used by "countless mathematicians"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.

Death

Cohen and his wife, Christina (née Karls), had three sons. Cohen died on March 23, 2007, inStanford, California,after suffering fromlung disease.

Selected publications

• Cohen, Paul Joseph (1958)."Topics in the theory of uniqueness of trigonometrical series"(PDF).Archived fromthe original(PDF)on 2011-07-25.Retrieved2010-02-19.
• Cohen, Paul Joseph (1960). "On a conjecture of Littlewood and idempotent measures".Amer. J. Math.82(2): 191–212.doi:10.2307/2372731.JSTOR2372731.MR0133397.
• Cohen, Paul Joseph (December 1963)."The independence of the continuum hypothesis".Proceedings of the National Academy of Sciences of the United States of America.50(6): 1143–1148.Bibcode:1963PNAS...50.1143C.doi:10.1073/pnas.50.6.1143.PMC221287.PMID16578557.
• Cohen, Paul Joseph (January 1964)."The independence of the continuum hypothesis, II".Proceedings of the National Academy of Sciences of the United States of America.51(1): 105–110.Bibcode:1964PNAS...51..105C.doi:10.1073/pnas.51.1.105.PMC300611.PMID16591132.
• Cohen, Paul Joseph (2008) [1966].Set theory and the continuum hypothesis.Mineola, New York City: Dover Publications. p. 151.ISBN978-0-486-46921-8.

See also

• Cohen algebra

Further reading

• Akihiro Kanamori,"Cohen and Set Theory",The Bulletin of Symbolic Logic,Volume 14, Number 3, Sept. 2008.
• Sarnak, Peter(December 2007)."Remembering Paul Cohen"(PDF).MAA Focus.27(9). Washington, DC: Mathematical Association of America: 21–22.ISSN0731-2040.Retrieved2009-05-31.

External links

• O'Connor, John J.;Robertson, Edmund F.,"Paul Joseph Cohen",MacTutor History of Mathematics Archive,University of St Andrews
• Paul Joseph Cohenat theMathematics Genealogy Project
• paulcohen.org- a commemorative website celebrating the life of Paul Cohen
• Stanford obituary