It is believed that to disprove it as a theorem would require axioms that guarantee the existence of sets having cardinalities different than those proven to exist.  
In 1963 Paul Cohen introduced the method of forcing, in his proof that this concept's generalized version implies its original formulation.  Combining a proof by G?del with Cohen's proof shows that it is independent of the axiom of choice.  It is believed that to disprove it as a theorem would require axioms that guarantee the existence of sets having cardinalities different than those proven to exist.  For 10 points, name this conjecture that can be stated as: there is no cardinality between an infinite set's cardinality and that of its power set.
In 1963 Paul Cohen introduced the method of forcing, in his proof that this concept's generalized version implies its original formulation.  
continuum hypothesis