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# Paul Cohen

Paul Joseph Cohen (born April 2, 1934) is an American mathematician. He was born in Long Branch, New Jersey and graduated in 1950 from Stuyvesant High School in New York City.

He then studied at Brooklyn College for his bachelor's degree of 1953. At the University of Chicago, he received his master's degree in 1954 and his PhD in 1958.

He is noted for inventing a technique called forcing which he used to show that neither the continuum hypothesis nor the axiom of choice can be proved from the standard Zermelo-Fraenkel axioms of set theory. In conjunction with the earlier work of Gödel, this showed that both these statements are independent of the Zermelo-Fraenkel axioms: they can be neither proved nor disproved from these axioms. For his efforts he won the Fields Medal in 1966. He was also awarded the Bôcher Memorial Prize in 1964 for his work in mathematical analysis.

This result is possibly the most famous non-trivial example illustrating the incompleteness of a formal system.

On the Continuum Hypothesis

"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 $\aleph_1$ is 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 $\aleph_n, \aleph_\omega, \aleph_a$, where $a = \aleph_\omega$, 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. "

• Cohen, P. Set Theory and the Continuum Hypothesis p.151.