Effectively closed sets and enumerations

Paul Brodhead, Douglas A. Cenzer

Abstract: An effectively closed set, or \({\Pi^{0}_{1}}\) class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of \({\Pi^{0}_{1}}\) classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of \({\Pi^{0}_{1}}\) classes and for the subclasses of decidable and of homogeneous \({\Pi^{0}_{1}}\) classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends.