Many examples comes to mind, the most famous being the Gödel's theorems viewed as formalisations of the Liar's paradox. I just realised that the proof of non-calculability of Kolmogorov complexity is a positive rewriting of Berry's paradox. My question (perhaps to be made into collective mode) is a) what are the best examples you know ? b) (more important) is there some explication of this productivity of paradoxes (or, conversely, do you know of paradoxes with no interesting follow-up) ?