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Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 565–582 | Cite as

Effectively closed sets and enumerations

  • Paul Brodhead
  • Douglas Cenzer
Article

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.

Mathematics Subject Classification (2000)

03D30 03D25 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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