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Paul J. Cohen

Paul J. Cohen
Born	April 2, 1934 Long Branch, New Jersey
Died	March 23, 2007 Stanford, California
Field	Mathematics
Institution	Stanford University
Alma mater	University of Chicago
Academic advisor	Antoni Zygmund
Notable students	Peter Sarnak
Known for	Forcing Continuum hypothesis
Notable prizes	Fields Medal (1966) Bôcher Prize (1964) National Medal of Science (1967)

Paul Joseph Cohen (April 2, 1934 – March 23, 2007^[1]) was an American mathematician. He was born in Long Branch, New Jersey into a Jewish family and graduated in 1950 from Stuyvesant High School in New York City. Image File history File links Current_event_marker. ... Wikinews has news related to: Obituaries // The following is a list of notable deaths in 2007. ... Image File history File links No higher resolution available. ... April 2 is the 92nd day of the year (93rd in leap years) in the Gregorian calendar, with 273 days remaining. ... 1934 (MCMXXXIV) was a common year starting on Monday (link will take you to calendar). ... Map of Long Branch in Monmouth County Long Branch is a City located in Monmouth County, New Jersey. ... March 23 is the 82nd day of the year in the Gregorian calendar (83rd in leap years). ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ... Stanford is a census-designated place (CDP) located in Santa Clara County, California. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The Leland Stanford Junior University, commonly known as Stanford University (or simply Stanford), is a private university located approximately 37 miles (60 kilometers) southeast of San Francisco and approximately 20 miles northwest of San José in an unincorporated area of Santa Clara County. ... The University of Chicago is a private university located principally in the Hyde Park neighborhood of Chicago. ... Antoni Zygmund (25 December 1900 _ 30 May 1992) was a Polish mathematician. ... Peter Sarnak (born 18 December 1953, Johannesburg) is a South African-born mathematician. ... In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). ... National Medal of Science The National Medal of Science, also called the Presidential Medal of Science, is an honor given by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social... April 2 is the 92nd day of the year (93rd in leap years) in the Gregorian calendar, with 273 days remaining. ... 1934 (MCMXXXIV) was a common year starting on Monday (link will take you to calendar). ... March 23 is the 82nd day of the year in the Gregorian calendar (83rd in leap years). ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ... Leonhard Euler is considered by many to be one of the greatest mathematicians of all time A mathematician is the person whose primary area of study and research is the field of mathematics. ... Map of Long Branch in Monmouth County Long Branch is a City located in Monmouth County, New Jersey. ... The word Jew ( Hebrew: יהודי) is used in a wide number of ways, but generally refers to a follower of the Jewish faith, a child of a Jewish mother, or someone of Jewish descent with a connection to Jewish culture or ethnicity and often a combination... 1950 (MCML) was a common year starting on Sunday (link will take you to calendar). ... Stuyvesant High School, affectionately known as Stuy, is a New York City public high school that specializes in mathematics and science. ... Midtown Manhattan, looking north from the Empire State Building, 2005 New York City (officially named the City of New York) is the most populous city in the state of New York and the entire United States. ...

He then studied at Brooklyn College from 1950 to 1953 but left before receiving a bachelor's degree when he learned he could pursue graduate studies in Chicago with just two years of college under his belt. At the University of Chicago, he received his master's degree in 1954 and his PhD in 1958 under supervision of Antoni Zygmund. Brooklyn College is a senior college of the City University of New York, located in Brooklyn, New York. ... 1950 (MCML) was a common year starting on Sunday (link will take you to calendar). ... 1953 (MCMLIII) was a common year starting on Thursday. ... A bachelors degree (Artium Baccalaureus, A.B. or B.A.) is usually an undergraduate academic degree awarded for a course or major that generally lasts for three, four, or in some cases and countries, five or six years. ... Nickname: The Windy City, The Second City, Chi Town, City of the Big Shoulders, The 312, The City that Works. Motto: Urbs In Horto (Latin: City in a Garden), I Will Location in Chicagoland and Illinois Coordinates: Country United States State Illinois County Cook & DuPage Incorporated March 4, 1837 Government... The University of Chicago is a private university located principally in the Hyde Park neighborhood of Chicago. ... A masters degree is an academic degree usually awarded for completion of a postgraduate (or graduate) course of one to three years in duration. ... 1954 (MCMLIV) was a common year starting on Friday of the Gregorian calendar. ... PhD usually refers to the academic title Doctor of Philosophy PhD can also refer to the manga Phantasy Degree This is a disambiguation page — a list of pages that otherwise might share the same title. ... Year 1958 (MCMLVIII) was a common year starting on Wednesday of the Gregorian calendar. ... Antoni Zygmund (25 December 1900 _ 30 May 1992) was a Polish mathematician. ...

He is noted for inventing a technique called forcing which he used to show 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 these statements are independent of ZF: they can be neither proved nor disproved from these axioms. In this sense CH is undecidable, and probably the most famous example of a natural statement independent from the conventional axioms of set theory. In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... [...]I dont believe in natural science. ...

For his result on CH he won the Fields Medal in 1966 and the National Medal of Science in 1967. The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... National Medal of Science The National Medal of Science, also called the Presidential Medal of Science, is an honor given by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social... 1967 (MCMLXVII) was a common year starting on Sunday of the Gregorian calendar (the link is to a full 1967 calendar). ...

He was also awarded the Bôcher Memorial Prize in 1964 for his paper "On a conjecture of Littlewood and idempotent measures". The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). ... 1964 (MCMLXIV) was a leap year starting on Wednesday (the link is to a full 1964 calendar). ... In mathematics, the Littlewood conjecture is a open problem (as of 2004) in Diophantine approximation, posed by J. E. Littlewood around 1930. ...

He was a professor at Stanford University, where he supervised Peter Sarnak's dissertation, among others. The Leland Stanford Junior University, commonly known as Stanford University (or simply Stanford), is a private university located approximately 37 miles (60 kilometers) southeast of San Francisco and approximately 20 miles northwest of San José in an unincorporated area of Santa Clara County. ... Peter Sarnak (born 18 December 1953, Johannesburg) is a South African-born mathematician. ...

Angus MacIntyre of the University of London is reported as saying: "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 Godel, saying: "Nothing more dramatic than their work has happened in the history of the subject."^[2] The University of London is a university based primarily in London. ... Kurt Gödel Kurt Gödel [ kurt gøːdl ], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics, whose biography lists quite a few nations, although he is usually associated with Austria. ...

His twin sons Steven and Eric played the Dancing Twins on the TV show Ally McBeal.^[3] Ally McBeal is an American television series which ran on the FOX network from 1997 to 2002, and was one of the best-known dramedy television series of the 1990s. ...

On the continuum hypothesis

When studying the hypothesis, Cohen is quoted as saying that he "had the feeling that people thought the problem was hopeless since there was no new way of constructing models of set theory. Indeed,” he said in a 1985 interview, "they thought you had to be slightly crazy even to think about the problem." ^[4]

"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. 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 C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power 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 the replacement axiom can ever reach C. Thus C is greater than , where , 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."^[5] In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ... In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...

An "enduring and powerful product" of Cohen's work on the Hypothesis, and one used by "countless mathematicians" ^[4] is known as forcing and is used to construct mathematical models to test a given hypothesis for truth or falsehood. In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ...

Footnotes

1. ^ A public announcement of his death has been made by the American Mathematical Society [1].
2. ^ San Francisco Chronicle Obituary [2]
3. ^ dancingtwins.com - web site of Eric & Steve Cohen
4. ^ ^{a b} New York Times obituary, [3]
5. ^ Cohen, P. Set Theory and the Continuum Hypothesis p.151.

External links

• O'Connor, John J., and Edmund F. Robertson. "Paul Joseph Cohen". MacTutor History of Mathematics archive.
• Paul Joseph Cohen at the Mathematics Genealogy Project
• paulcohen.org - a commemorative website celebrating the life of Paul Cohen
• Stanford obituary

The Mathematics Genealogy Project is a web-based database that gives an academic genealogy based on dissertation supervision relations. ...