Paul Cohen (mathematician)
Encyclopedia
Paul Joseph Cohen was an American
mathematician
best known for his proof of the independence
of the continuum hypothesis
and the axiom of choice from Zermelo–Fraenkel set theory
, the most widely accepted axiomatization of set theory
.
, into a Jewish family that had immigrated to the U.S. from Poland. He graduated in 1950 from Stuyvesant High School
in New York City
.
Cohen next studied at the Brooklyn College
from 1950 to 1953, but he left before earning his bachelor's degree
when he learned that he could start his graduate studies at the University of Chicago
with just two years of college. At Chicago
, Cohen completed his master's degree
in mathematics
in 1954 and his Doctor of Philosophy
degree in 1958, under supervision of the Professor of Mathematics, Antoni Zygmund
. The subject of his doctoral thesis was Topics in the Theory of Uniqueness of Trigonometric Series.
, which he used to prove 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 of these statements are logically 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 probably 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
in mathematics in 1966, and also the National Medal of Science
in 1967. The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic.
Apart from his work in set theory, Cohen also made many valuable contributions to Analysis. He was awarded the Bôcher Memorial Prize
in mathematical analysis
in 1964 for his paper "On a conjecture by Littlewood
and idempotent
measures
", and lends his name to the Cohen-Hewitt factorization theorem.
Cohen was a professor at Stanford University
, where he supervised Peter Sarnak
's graduate research, among those of other students.
Angus MacIntyre of the University of London
stated about Cohen: "He was dauntingly clever, and one would have had to be naïve 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." 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."
"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 
[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
.
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" 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 problem of the Continuum Hypothesis at the Gödel centennial conference, in Vienna 2006. A video of this lecture is now available online.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
best known for his proof of the independence
Independence (mathematical logic)
In mathematical logic, independence refers to 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, and it is also impossible to prove from T that...
of the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
and the axiom of choice from Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
, the most widely accepted axiomatization of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
.
Early years
Cohen was born in Long Branch, New JerseyLong Branch, New Jersey
Long Branch is a city in Monmouth County, New Jersey, United States. As of the 2010 United States Census, the city population was 30,719.Long Branch was formed on April 11, 1867, as the Long Branch Commission, from portions of Ocean Township...
, into a Jewish family that had immigrated to the U.S. from Poland. He graduated in 1950 from Stuyvesant High School
Stuyvesant High School
Stuyvesant High School , commonly referred to as Stuy , is a New York City public high school that specializes in mathematics and science. The school opened in 1904 on Manhattan's East Side and moved to a new building in Battery Park City in 1992. Stuyvesant is noted for its strong academic...
in New York City
New York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
.
Cohen next studied at the Brooklyn College
Brooklyn College
Brooklyn College is a senior college of the City University of New York, located in Brooklyn, New York, United States.Established in 1930 by the New York City Board of Higher Education, the College had its beginnings as the Downtown Brooklyn branches of Hunter College and the City College of New...
from 1950 to 1953, but he left before earning his bachelor's degree
Bachelor's degree
A bachelor's degree is usually an academic degree awarded for an undergraduate course or major that generally lasts for three or four years, but can range anywhere from two to six years depending on the region of the world...
when he learned that he could start his graduate studies at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
with just two years of college. At Chicago
Chicago
Chicago is the largest city in the US state of Illinois. With nearly 2.7 million residents, it is the most populous city in the Midwestern United States and the third most populous in the US, after New York City and Los Angeles...
, Cohen completed his master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
in 1954 and his Doctor of Philosophy
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
degree in 1958, under supervision of the Professor of Mathematics, Antoni Zygmund
Antoni Zygmund
Antoni Zygmund was a Polish-born American mathematician.-Life:Born in Warsaw, Zygmund obtained his PhD from Warsaw University and became a professor at Stefan Batory University at Wilno...
. The subject of his doctoral thesis was Topics in the Theory of Uniqueness of Trigonometric Series.
Contributions to mathematics
Cohen is noted for developing a mathematical technique called forcingForcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
, which he used to prove that neither the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
(CH), nor the axiom of choice, can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. In conjunction with the earlier work of Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, this showed that both of these statements are logically 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 probably 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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, 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...
in mathematics in 1966, and also the National Medal of Science
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 sciences, biology, chemistry, engineering, mathematics and...
in 1967. The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic.
Apart from his work in set theory, Cohen also made many valuable contributions to Analysis. He was awarded the Bôcher Memorial Prize
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 . It is awarded every five years for a notable research memoir in analysis that has appeared during the past six years in a recognized North American...
in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
in 1964 for his paper "On a conjecture by Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
and idempotent
Idempotence
Idempotence is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application...
measures
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
", and lends his name to the Cohen-Hewitt factorization theorem.
Cohen was a professor at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
, where he supervised Peter Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
's graduate research, among those of other students.
Angus MacIntyre of the University of London
University of London
-20th century:Shortly after 6 Burlington Gardens was vacated, the University went through a period of rapid expansion. Bedford College, Royal Holloway and the London School of Economics all joined in 1900, Regent's Park College, which had affiliated in 1841 became an official divinity school of the...
stated about Cohen: "He was dauntingly clever, and one would have had to be naïve 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 Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, 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."
On the Continuum Hypothesis
While studying the continuum 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 an interview in 1985, "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 the axiom of infinity
Axiom of infinity
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...
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 the power set axiomAxiom of power set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:...
. 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
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set...
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" is known as forcing
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
, 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 problem of the Continuum Hypothesis at the Gödel centennial conference, in Vienna 2006. A video of this lecture is now available online.
Further reading
- Akihiro KanamoriAkihiro Kanamoriis a Japan-born American mathematician. He specializes in set theory and is the author of the successful monograph on large cardinals, The Higher Infinite. He wrote several essays on the history of mathematics, especially set theory.Kanamori graduated from California Institute of Technology and...
, "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
- Stanford obituary