Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American

Cohen and Set Theory

, ''The Bulletin of Symbolic Logic'', Volume 14, Number 3, Sept. 2008. *

paulcohen.org

- a commemorative website celebrating the life of Paul Cohen

{{DEFAULTSORT:Cohen, Paul 1934 births 2007 deaths 20th-century American mathematicians 21st-century American mathematicians American people of Polish-Jewish descent Brooklyn College alumni Fields Medalists Institute for Advanced Study visiting scholars Members of the United States National Academy of Sciences National Medal of Science laureates People from Long Branch, New Jersey Set theorists Stuyvesant High School alumni University of Chicago alumni Mathematicians from New York (state) Members of the American Philosophical Society

mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
...

. He is best known for his proofs that the continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

and the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

are independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...

from Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...

, for which he was awarded a Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...

.
Early life and education

Cohen was born inLong Branch, New Jersey
Long Branch is a beachside city in Monmouth County, in the U.S. state of New Jersey. As of the 2010 U.S. census, the city's population was 30,719,Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""T ...

family that had immigrated to the United States from what is now Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It is divided into 16 administrative provinces called voivodeships, covering an area of . Poland has a population of over 38 million and is the fifth-most populou ...

; he grew up in Brooklyn
Brooklyn () is a borough of New York City, coextensive with Kings County, in the U.S. state of New York. Kings County is the most populous county in the State of New York, and the second-most densely populated county in the United States, beh ...

.. He graduated in 1950, at age 16, from Stuyvesant High School
, motto_translation = For knowledge and wisdom
, address = 345 Chambers Street
, city = New York
, state = New York
, zipcode = 10282
, country ...

in New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the L ...

.
Cohen next studied at the Brooklyn College
, mottoeng = Nothing without great effort
, established =
, parent = CUNY
, type = Public university
, endowment = $98.0 million (2019)
, budget = $123.96 m ...

from 1950 to 1953, but he left without earning his bachelor's degree
A bachelor's degree (from Middle Latin ''baccalaureus'') or baccalaureate (from Modern Latin ''baccalaureatus'') is an undergraduate academic degree awarded by colleges and universities upon completion of a course of study lasting three to six ...

when he learned that he could start his graduate studies at the University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the ...

with just two years of college. At Chicago
(''City in a Garden''); I Will
, image_map =
, map_caption = Interactive Map of Chicago
, coordinates =
, coordinates_footnotes =
, subdivision_type = Country
, subdivision_name ...

, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...

degree in 1958, under supervision of Antoni Zygmund
Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century ...

. The title of his doctoral thesis was ''Topics in the Theory of Uniqueness of Trigonometrical Series''.
Cohen was a member of the American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...

, the United States National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Na ...

, and the American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

. On June 2, 1995 Cohen received an honorary doctorate
An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or ''ad hono ...

from the Faculty of Science and Technology at Uppsala University
Uppsala University ( sv, Uppsala universitet) is a public research university in Uppsala, Sweden. Founded in 1477, it is the oldest university in Sweden and the Nordic countries still in operation.
The university rose to significance during ...

, Sweden
Sweden, formally the Kingdom of Sweden,The United Nations Group of Experts on Geographical Names states that the country's formal name is the Kingdom of SwedenUNGEGN World Geographical Names, Sweden./ref> is a Nordic countries, Nordic c ...

.
Career

Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither thecontinuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

(CH) nor the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent In mathematical logic, independence is the unprovability of a sentence from other sentences.
A sentence σ is independent of a given first-order theory ''T'' if ''T'' neither proves nor refutes σ; that is, it is impossible to prove σ from ''T ...

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 the Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...

in mathematics in 1966, and also the National Medal of Science
The National Medal of Science is an honor bestowed 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 scienc ...

in 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 the Bôcher Memorial Prize
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). It is awarded every three years (formerly every five yea ...

in mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...

in 1964 for his paper "On a conjecture by Littlewood and idempotent measures", and lends his name to the Cohen–Hewitt factorization theorem In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

.
Cohen was a full professor of mathematics at Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is conside ...

. He was an Invited Speaker at the ICM in 1962 in Stockholm and in 1966 in Moscow.
Angus MacIntyre
Angus John Macintyre FRS, FRSE (born 1941) is a British mathematician and logician who is a leading figure in model theory, logic, and their applications in algebra, algebraic geometry, and number theory. He is Emeritus Professor of Mathematics ...

of the Queen Mary University of London
Queen Mary University of London (QMUL, or informally QM, and previously Queen Mary and Westfield College) is a public university, public research university in Mile End, East London, England. It is a member institution of the federal University of ...

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
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

, 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 pendenceof the cont nuumhypthesis
A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...

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 ohenfeels may eventually come to be accepted is that CH is obviously false. The main reason one accepts theaxiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...

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 $\backslash aleph\_1$ 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 $C$ he continuumis, 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
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinit ...

can ever reach $C$.
Thus $C$ is greater than $\backslash aleph\_n,\; \backslash aleph\_\backslash omega,\; \backslash aleph\_a$, where $a\; =\; \backslash aleph\_\backslash omega$, 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."
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 in Vienna
en, Viennese
, iso_code = AT-9
, registration_plate = W
, postal_code_type = Postal code
, postal_code =
, timezone = CET
, utc_offset = +1
, timezone_DST ...

.
Death

Cohen and his wife, Christina (née Karls), had three sons. Cohen died on March 23, 2007 inStanford, California
Stanford is a census-designated place (CDP) in the northwest corner of Santa Clara County, California, United States. It is the home of Stanford University. The population was 21,150 at the 2020 census.
Stanford is an unincorporated area of ...

after suffering from lung disease
The lungs are the primary organs of the respiratory system in humans and most other animals, including some snails and a small number of fish. In mammals and most other vertebrates, two lungs are located near the backbone on either side of ...

.
Selected publications

* *See also

* Cohen algebraReferences

Further reading

*Akihiro Kanamori
is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinals, ''The Higher Infinite''. He has written several essays on the history of mathematics, especially set theory.
Kanamori ...

,Cohen and Set Theory

, ''The Bulletin of Symbolic Logic'', Volume 14, Number 3, Sept. 2008. *

External links

* *paulcohen.org

- a commemorative website celebrating the life of Paul Cohen

{{DEFAULTSORT:Cohen, Paul 1934 births 2007 deaths 20th-century American mathematicians 21st-century American mathematicians American people of Polish-Jewish descent Brooklyn College alumni Fields Medalists Institute for Advanced Study visiting scholars Members of the United States National Academy of Sciences National Medal of Science laureates People from Long Branch, New Jersey Set theorists Stuyvesant High School alumni University of Chicago alumni Mathematicians from New York (state) Members of the American Philosophical Society