Continuum hypothesis was proven, with an proving method called
"forcing", to be undecidable under commonly accepted axioms of the
set theory. This means that neither continuum hypothesis nor it's
negation follows from this axioms just like one axiom (or it's
negation) in some consistent axiomatic system does not follow from
other axioms. Therefore, continuum hypothesis or it's negation
could be added as an additional axiom to existing commonly accepted
axioms of the set theory.