Recursively enumerable generic sets

Abstract We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0′. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice ℰ of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions.