Enumerations of [formula omitted] Classes: Acceptability and Decidable Classes

Abstract

A Π10 class is an effectively closed set of reals. One way to view it is as the set of infinite paths through a computable tree. We consider the notion of acceptably equivalent numberings of Π10 classes. We show that a permutation exists between any two acceptably equivalent numberings that preserves the computable content. Furthermore the most commonly used numberings of the Π10 classes are acceptably equivalent. We also consider decidable Π10 classes in enumerations. A decidable Π10 class may be represented by a unique computable tree without dead ends, but we show that this tree may not show up in an enumeration of uniformly computable trees which gives rise to all Π10 classes. In fact this is guaranteed to occur for some decidable Π10 class. These results are motivated by structural questions concerning the upper semilattice of enumerations of Π10 classes where notions such as acceptable equivalence arise.