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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.
This result is possibly the most famous non-trivial example illustrating Gödel's incompleteness theorem.
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 aleph1 is the set of countable ordinals and this is merely a special and the simplest way of generating a highter cardinal. The set C 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 alephn, alephw, alepha, where a= alephw, 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. "