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Bar recursion versus polymorphism

By E. Barendsen and M.A. Bezem


By constructing a counter, model we show that a certain appealing\ud equation E has no solution in Girard's [1972] second order lambdacalculus\ud (the so-called polymorphic, lambda calculus). The equation E = Eφ (with φ a type 3 variable) is a simple functional equation in thelkiguage\ud of Gödel's [1958] system of higher order primitive recursive fanctiotals and\ud has an easy solution in Spector's [1962] system of bar recursive functionals.\ud This shows that the class of bar recursive functionals differs from the class \ud of functionals definable in the polymorphic lambda calculus. The fact\ud that the two calculi have different classes of definable functionals (at least\ud of type 3), contrasts the metamathematical results from Spector [1962]\ud and Girard [1972] which state that the two calculi have the same class\ud of definable functions, namely the provably total recursive functions of analysis

Topics: Wijsbegeerte
Year: 1992
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