Graham Priest's Logic of Paradox is a modification of classical logic where the principle of explosion does not hold, so that there are inconsistent theories which are not automatically trivial. Priest writes that in general, we may use classical logic as long as no paradoxial sentences are involved. But sometimes it might turn out that a part of the proof was based on a paradox so that it turns out to be invalid (because the proof used rules of inference that apply only to non-paradoxical sentences, sentences that are either only true or only false but not both).
What are interesting examples in the history of mathematics where a proof relied on a paradoxical proposition? Could it be in principle any proof that turned out false?
"However, it is always possiile that semantic terms may be smuggled implicitly into a sentence without our knowledge or that paradoxicality arises for some other reason of which we are not aware. We may then invoke the maxim but find that at a later time evidence turns up to the effect that there are paradoxical sentences in the proof and that they occur in such a way as to invalidate a quasi-valid rule of inference. We will then have to reject the proof that we previously accepted. Although this may sound a little unrealistic, there is in fact nothing essentially new in this situation to those who are familiar with a little of the history of mathematics." http://philpapers.org/rec/PRITLO