Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent).  Would it be possible to have a zero-knowledge proof of this?  In other words, would it be possible for me to convince you with high probability that I had derived such a proof without (feasibly) revealing the proof of the contradiction?  (I haven't by the way...found such a contradiction.)