Earlier quoted context omitted.
There's no obstacle in principle to proving that something can't be proved. As far as I understand things, the continuum hypothesis has been shown to be unprovable within ZFC (since ZFC is consistent with both the truth and the falsity of the continuum hypothesis, obviously it is unable to prove the hypothesis true or false), and theoretical computer science produces results of the form "to prove X, the following kin…
"ZFC is consistent with …" must, of course, be replaced by "If ZFC is consistent, then so are …". In fact, singling out ZFC gives you another example of this kind of result: if ZF is consistent, then so are ZF+C and ZF-C.
That all depends whether you're comfortable saying a contradiction is "consistent" with an unrelated proposition; I don't see much of a problem with it. Contradictions, being impossible, are logically consistent with everything else. I guess unprovable propositions would switch over to being provable, though.