Preprint-reihe Mathematik Ernst-moritz-arndt-universit at Greifswald Approximate Decidability in Euclidean Spaces { Draft { Approximate Decidability in Euclidean Spaces { Draft {

We study concepts of decidability (recursivity) for subsets of Euclidean spaces R k within the framework of approximate computability (type two theory of ef-fectivity). A new notion of approximate decidability is proposed and discussed in some detail. It is an eeective variant of F. Hausdorr's concept of resolvable sets and modiies and generalizes notions of recursivity known from computable analysis, formerly used for open or closed sets only, to more general types of sets. Approximate decidability of a set can equivalently be expressed by computability of the characteristic function by means of an appropriately working oracle Turing machine. The notion fulllls some further natural requirements and is invariant under canonical embeddings of sets into spaces of higher dimensions. However, it is not closed under binary union or intersection of sets. We also show how the framework of resolvability and approximate decidabil-ity can be applied to investigate concepts of reducibility for subsets of Euclidean spaces.