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: Kurt Gödel showed in 1939 , that the hypothesis can not be falsified using Zermelo -- Fraenkel set theory .