Gödel proved in 1938 that if the axioms of ZF set theory are consistent, then they are also consistent with the continuum hypothesis and the axiom of choice. Thus, one cannot expect to refute these principles, and in this sense, it is safe to assume that they are true. This result explains why Cantor was not able to find a definitive set of intermediate cardinality between ℕ and ℝ, since it is consistent with set theory that there is no such set.
