Cohen, who was born at Long Branch, New Jersey, was educated at Brooklyn College and at the University of Chicago, where he obtained his PhD in 1958. He spent a year at the Massachusetts Institute of Technology and two years at the Institute for Advanced Studies, Princeton, before moving to Stanford in 1961. He was appointed professor of mathematics in 1964.

Mathematicians had been introduced to transfinite arithmetic by Georg Cantor from the 1870s onwards. Cantor had identified two distinct infinite sets, namely the set of natural numbers and the set of real numbers, represented by ℵ₀ and c respectively. He had also proved that there were an infinite number of infinite numbers, that following ℵ₀, there came ℵ₁, ℵ₂, ℵ₃… indefinitely. Where did c fit into this sequence? Cantor answered by proposing that c = ℵ₁, a supposition since known as the ‘continuum hypothesis’. It was the first member on David Hilbert's 1900 list of outstanding unsolved mathematical problems.

Little progress was made upon the problem before 1938 when Kurt Gödel demonstrated that set theory remains consistent if the continuum hypothesis is added as an axiom. This did not, however, constitute a proof of the hypothesis, for set theory's own absolute consistency has never been proved. Nonetheless, Gödel's work did show that the continuum hypothesis could not be shown to be false within set theory.

In 1963 Cohen proposed to develop a non-Cantorian set theory that contained not the continuum hypothesis but its negation. He showed that no contradiction ensued and it seemed to follow that the continuum hypothesis was quite independent of set theory and that it could be neither proved nor disproved within any standard system of set theory.