Original: The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo -- Fraenkel set theory , the standard foundation of modern mathematics , provided set theory is consistent .
Simplified: Paul Cohen showed in the 1960s that the Zermelo-Fraenkel set theory can not be use to prove the continuum hypothesis , either .