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Effectively closed sets and enumerations. (English) Zbl 1140.03026
Summary: An effectively closed set, or \({\Pi^{0}_{1}}\) class, may be viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from \(\omega\) 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.

03D45 Theory of numerations, effectively presented structures
03D25 Recursively (computably) enumerable sets and degrees
03D30 Other degrees and reducibilities in computability and recursion theory
Full Text: DOI
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