Original: The contributions of Kurt GÃ del in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermeloâ '' Fraenkel set theory , the standard foundation of modern mathematics , provided ZF set theory is consistent .
Simplified: Kurt GÃ del showed in 1939 , that the hypothesis can not be falsified using Zermeloâ '' Fraenkel set theory . The Zermeloâ '' Fraenkel set theory is the set theory commonly used in mathematics .