Paul Cohen

Paul Joseph Cohen (April 2, 1934 – March 23, 2007^[1][2]) 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.

Early years

Paul J. Cohen was born in Long Branch, New Jersey into a Jewish family. He graduated in 1950 from Stuyvesant High School in New York City.^[2]

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. At the University of Chicago, he received his master's degree in 1954 and his PhD in 1958 under supervision of Antoni Zygmund. His doctoral thesis was Topics in the Theory of Uniqueness of Trigonometric Series.^[3]

Contributions to mathematics

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.

For his result on CH he won the Fields Medal in 1966 and the National Medal of Science in 1967; the 1966 Fields Medal continues to be the only Fields Medal to have been awarded for a work in mathematical logic.

He was also awarded the Bôcher Memorial Prize in analysis in 1964 for his paper "On a conjecture of Littlewood and idempotent measures".

He was a professor at Stanford University, where he supervised Peter Sarnak's dissertation, among others.

Angus MacIntyre of the University of London is reported as saying: "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."^[4] Gödel himself wrote a letter to Cohen in 1963, a draft of which states "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."^[5]

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."^[6]

"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."^[7]

An "enduring and powerful product" of Cohen's work on the Hypothesis, and one used by "countless mathematicians" ^[6] is known as forcing and is used to construct mathematical models to test a given hypothesis for truth or falsehood.

Shortly before his death, Paul Cohen gave a fascinating lecture describing his solution to the Continuum Hypothesis at the Gödel centennial conference, Vienna 2006. A video of this lecture is now available online.^[8]

Notes

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 (Washington, DC: Mathematical Association of America) 27 (9): 21–22. ISSN 0731-2040. http://www.math.princeton.edu/sarnak/RememberingPaulCohen.pdf. Retrieved 2009-05-31. 

External links

• O'Connor, John J.; Robertson, Edmund F., "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