Paul Cohen

Last updated • 6 min readFrom Wikipedia, The Free Encyclopedia
Paul J. Cohen
Born(1934-04-02)April 2, 1934
Long Branch, New Jersey, U.S.
DiedMarch 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) [1] was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. [2]

Contents

Early life and education

Cohen was born in Long Branch, New Jersey in 1934, into a Jewish family that had immigrated to the United States from what is now Poland; he grew up in Brooklyn. [3] [4] He graduated in 1950, at age 16, from Stuyvesant High School in New York City. [1] [4]

Cohen next studied at the Brooklyn College from 1950 to 1953, but he left without 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 Antoni Zygmund. The title of his doctoral thesis was Topics 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 the University of Rochester for a year. He then spent the academic year 1958–59 at the Massachusetts Institute of Technology before spending 1959–61 as a fellow at the Institute for Advanced Study at Princeton. These were years in which Cohen made a number of significant mathematical breakthroughs. In Factorization 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 by Walter Rudin. In Cohen (1960) he made a significant breakthrough in solving the Littlewood conjecture. [7]

Cohen was a member of the American Academy of Arts and Sciences, [8] the United States National Academy of Sciences, [9] and the American Philosophical Society. [10] On June 2, 1995, Cohen received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden. [11]

Career

Cohen is noted for developing a mathematical technique called forcing, 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 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. [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 the Bôcher Memorial Prize in mathematical analysis in 1964 for his paper "On a conjecture by Littlewood and idempotent measures", [13] and lends his name to the Cohen–Hewitt factorization theorem.

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

Angus MacIntyre of the Queen 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 to Kurt 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]

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." [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 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.

—  Cohen (2008)

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 in Vienna. [17]

Death

Cohen and his wife, Christina (née Karls), had three sons. Cohen died on March 23, 2007, in Stanford, California, after suffering from lung disease. [18]

Selected publications

See also

Related Research Articles

In mathematics, specifically set theory, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

<span class="mw-page-title-main">Cardinal number</span> Size of a possibly infinite set

In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter (aleph) marked with subscript indicating their rank among the infinite cardinals.

<span class="mw-page-title-main">Cardinality</span> Definition of the number of elements in a set

In mathematics, cardinality describes a relationship between sets which compares their relative size. For example, the sets and are the same size as they each contain 3 elements. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two notions often used when referring to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set may also be called its size, when no confusion with other notions of size is possible.

Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

<span class="mw-page-title-main">Set theory</span> Branch of mathematics that studies sets

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 concerned with those that are relevant to mathematics as a whole.

In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers.

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 as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wacław Sierpiński.

<span class="mw-page-title-main">Aleph number</span> Infinite cardinal number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930.

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L. The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms. Generalizations of this axiom are explored in inner model theory.

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by or

<span class="mw-page-title-main">Axiom of limitation of size</span>

In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure . Just as the axiom of projective determinacy yields a canonical theory of , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.

<span class="mw-page-title-main">Menachem Magidor</span> Israeli mathematician

Menachem Magidor is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of the Association for Symbolic Logic from 1996 to 1998 and as president of the Division for Logic, Methodology and Philosophy of Science and Technology of the International Union for History and Philosophy of Science (DLMPST/IUHPS) from 2016 to 2019. In 2016 he was elected an honorary foreign member of the American Academy of Arts and Sciences. In 2018 he received the Solomon Bublick Award.

In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the power set of the natural numbers.

In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between , and the cardinality of the continuum, that is, the cardinality of the set of all real numbers. The latter cardinal is denoted or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them.

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that . It is the negation of a weakened form, , of the Continuum Hypothesis (CH). It was discussed by Nikolai Luzin in 1935, although he did not claim to be the first to postulate it. The statement may also be called Luzin's hypothesis.

The term weak continuum hypothesis can be used to refer to the hypothesis that , which is the negation of the second continuum hypothesis. It is equivalent to a weak form of ◊ on . F. Burton Jones proved that if it is true, then every separable normal Moore space is metrizable.

References

  1. 1 2 Levy, Dawn (2007-03-28). "Paul Cohen, winner of world's top mathematics prize, dies at 72". Stanford Report. Retrieved 2007-10-31.
  2. ↑ Pearce, Jeremy (2 April 2007). "Paul J. Cohen, Mathematics Trailblazer, Dies at 72". NY Times.
  3. ↑ Macintyre, A.J. "Paul Joseph Cohen" Archived 2010-12-25 at the Wayback Machine , London Mathematical Society. Accessed March 3, 2011. "Cohen's origins were humble. He was born in Long Branch, New Jersey on 2 April 1934, into a Polish immigrant family."
  4. 1 2 Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Paul Cohen", More Mathematical People, Harcourt Brace Jovanovich, pp. 42–58.
  5. ↑ Cohen 1958.
  6. ↑ "Topics in the Theory of Uniqueness of Trigonometrical Series". ProQuest. Retrieved 2024-11-02.
  7. ↑ O'Connor, John J.; Robertson, Edmund F., "Paul Joseph Cohen", MacTutor History of Mathematics Archive , University of St Andrews
  8. ↑ "Paul Joseph Cohen". American Academy of Arts & Sciences. Retrieved 2022-08-22.
  9. ↑ "Paul J. Cohen". www.nasonline.org. Retrieved 2022-08-22.
  10. ↑ "APS Member History". search.amphilsoc.org. Retrieved 2022-08-22.
  11. ↑ "Honorary doctorates - Uppsala University, Sweden". www.uu.se. Retrieved 21 March 2018.
  12. ↑ "The President's National Medal of Science: Recipient Details - NSF - National Science Foundation". www.nsf.gov. Retrieved 21 March 2018.
  13. ↑ Cohen 1960.
  14. ↑ Davidson, Keay (2007-03-30). "Paul Cohen -- Stanford professor, acclaimed mathematician". San Francisco Chronicle . Retrieved 2007-10-31.
  15. ↑ Solomon Feferman, The Gödel Editorial Project: A synopsis p. 11.
  16. 1 2 Pearce, Jeremy (2007-04-02). "Paul J. Cohen, Mathematics Trailblazer, Dies at 72". The New York Times . Retrieved 2007-10-31.
  17. ↑ Paul Cohen lecture video, six parts, Gödel Centennial, Vienna 2006 on YouTube
  18. ↑ Pearce, Jeremy (2007-04-02). "Paul J. Cohen, Mathematics Trailblazer, Dies at 72". The New York Times. ISSN   0362-4331 . Retrieved 2020-06-13.

Further reading

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.