The Mathematician Who Changed What It Means to Solve a Problem

Paul Cohen entered mathematics as an analyst, then attacked a foundational problem that had resisted Cantor, Hilbert and generations of logicians. Gödel had shown that the continuum hypothesis could not be disproved from standard set theory; Cohen invented forcing to show it could not be proved either. His 1963 work established the hypothesis’s independence from ZFC, assuming consistency, and also yielded the independence of the axiom of choice from ZF. The breakthrough brought him the 1966 Fields Medal.

Cohen's achievement did something stranger than solving the continuum hypothesis: it proved that the usual axioms cannot decide it. To reach that conclusion he invented forcing, a method for extending models of set theory while controlling which statements become true. Combined with Gödel's earlier work, the result revealed that mathematically coherent universes can disagree about fundamental questions of infinity. Forcing then became a central tool of modern set theory, generating independence results far beyond the.