TITLE: Positive results coming from paradoxes
QUESTION [7 upvotes]: 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) ?

REPLY [5 votes]: Zeno's paradox on Achilles and the tortoise is related to the formula for infinite geometric sums and more generally to the idea that infinite sums can lead to finite outcomes. 
It is quite remarkable how relevant 17th century calculus is to Zeno's three paradoxes. In fact, it looks that in a different universe these paradoxes could have started calculus. Terry Tao remarked on some post I made about it: "Zeno's arrow paradox can be reinterpreted in the light of the theory of differential equations that the equations of motion must be second-order in time rather than first-order, since one has to specify initial velocity in addition to initial position in order to have a well-posed system. So the arrow paradox may well be the earliest precursor of Newton's famous equation F=ma..."