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 noncalculability 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 followup) ?

Kritchman and Raz adapted the surprise examination paradox (a/k/a the unexpected hanging paradox) to prove Godel's second incompleteness theorem (extending the ideas underlying Chaitin's Berry'sparadoxinspired proof of Godel's first incompleteness theorem). I blogged on this here. 


If you accept that the BanachTarski result is a "paradox" (although not an antinomy), then it was a productive paradox insofar as it led von Neumann to consider finitely additive invariant measures, work which eventually blossomed into the study of amenable groups. 


Russell's paradox resulted in formulation of type theory and set theory. 


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 secondorder in time rather than firstorder, since one has to specify initial velocity in addition to initial position in order to have a wellposed system. So the arrow paradox may well be the earliest precursor of Newton's famous equation F=ma..." 


More in the spirit than the letter of the question, some of Alan Turing's work can be seen as applications of contradiction. The development of the Turing Machine and using it to resolve Hilbert's Entscheidungsproblem can be seen as a successful use of the Liar Paradox, and elsewhere I read a quote which suggested that Turing took a result from mathematical logic (from a contradiction one can prove anything) and used it to help build one of the codebreaking systems used in World War II. Gerhard "Ask Me About Me Unasking" Paseman, 2012.12.15 

