The Continuum hypothesis states that there is no no set whose cardinality lies between that of the Natural numbers and that of the Reals.
The Generalised Continuum hypothesis states that if A is any set, there is no set whose cardinality lies between the cardinality of A and the cardinality of the set of all subsets of A.
The Continuum hypothesis has been shown to be independent of the other set-theory axioms.