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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.

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