It is known from a result of Sierpinski that the generalized continuum hypothesis (GCH) implies the axiom of choice (AC). It is also known from the celebrated results of Cohen that AC is independent of ZF and that GCH is independent of ZFC. But suppose we start with the axioms of ZF and assume they are consistent, and then add both the negation of the axiom of choice and the continuum hypothesis (i.e., CH but not GCH).

Is it known whether the resulting system is consistent or inconsistent?