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 . 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 .