Paul Cohen

Paul Cohen (1934–2007) was an American mathematician who proved the independence of the Continuum Hypothesis and the Axiom of Choice from the standard axioms of ZFC set theory. His technique — forcing — is one of the most powerful and versatile tools in modern set theory, allowing the construction of models of ZFC where specific statements are true or false without violating consistency.

Cohen's 1963 proof settled a problem that had remained open since Cantor first posed it in 1878 and that Hilbert had listed as the first of his 23 problems in 1900. The independence of the Continuum Hypothesis means that ZFC does not determine whether there are cardinalities strictly between the naturals and the reals. The axioms are consistent with a minimal universe where the next infinity after ℵ₀ is the continuum, and with a maximal universe where there are infinitely many intermediate sizes.

Forcing works by starting with a base model of ZFC and adding a new generic set that satisfies specific properties while preserving the consistency of the original axioms. The new set is generic in the sense that it is constructed to avoid all definable properties of the ground model, making it a kind of controlled randomness. This technique transformed set theory from a field that asked what