is an impossibility theorem proving that no incentive system for a team of agents can satisfy all of three desirable criteria. In natural science In natural science, impossibility assertions (like other assertions) come to be widely accepted as overwhelmingly probable rather than considered proved to the point of being unchallengeable. The basis for this strong acceptance is a combination of extensive evidence of something not occurring, combined with an underlying theory, very successful in making predictions, whose assumptions lead logically to the conclusion that something is impossible. Two examples of widely accepted impossibilities in physics are perpetual motion machines, which violate the law of conservation of energy, and exceeding the speed of light, which violates the implications of special relativity. Another is the uncertainty principle of quantum mechanics, which asserts the impossibility of simultaneously knowing both the position and the momentum of a particle. There is also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined. See also List of unsolved problems in mathematicsSolutions of these problems are still being searched for. In contrast, the above problems are known to have no solution. Notes and references Bibliography G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford England, 1979, reprinted 2000 with General Index (first edition: 1938). The proofs that e and pi are transcendental are not trivial, but a mathematically adept reader will be able to wade through them. Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press, 1962, reprint of 2nd edition 1927, first edition 1913. Chap. 2.I. "The Vicious-Circle Principle" p. 37ff, and Chap. 2.VIII. "The Contradictions" p. 60ff. (and ). online version This is the epochal paper where Turing defines Turing machines and shows that it (as well as the Entscheidungsproblem) is unsolvable. Martin Davis, The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, Raven Press, New York, 1965. Turing's paper is #3 in this volume. Papers include those by Godel, Church, Rosser, Kleene, and Post. Martin Davis's chapter "What is a Computation" in Lynn Arthur Steen's Mathematics