The unsolvable problem that launched a revolution in set theory

Paul Cohen developed forcing in 1963 to prove the Continuum Hypothesis cannot be proved from ZFC axioms. He received the 1966 Fields Medal for this achievement. While Gödel constructed a smaller model where the hypothesis holds, Cohen extended models outward where it fails.

Georg Cantor formulated the Continuum Hypothesis after proving some infinite sets are larger than others. He showed the real numbers form a larger infinity than the natural numbers. Natural numbers constitute a countable infinity while real numbers form an uncountable continuum.

Kurt Gödel constructed a model of ZFC in 1938 where the Continuum Hypothesis holds true. Starting from an assumed ZFC model M, Gödel created a subset L called the constructible universe. This model demonstrated the hypothesis cannot be disproved from ZFC axioms.

Kurt Gödel proved in 1938 that the Continuum Hypothesis is consistent with ZFC axioms. Paul Cohen proved in 1963 it cannot be proved from those axioms, earning the 1966 Fields Medal. ZFC (Zermelo-Fraenkel with Choice) axioms form the standard foundation for mathematics.