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Effectively closed sets and enumerations

Effectively closed sets and enumerations 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Effectively closed sets and enumerations

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References (43)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Algebra; Mathematics, general; Mathematical Logic and Foundations
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-008-0065-7
Publisher site
See Article on Publisher Site

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.

Journal

Archive for Mathematical LogicSpringer Journals

Published: Apr 12, 2008

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