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The Hausdorff-Ershov Hierarchy in Euclidean Spaces

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Abstract

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝk which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta 2. This is based on suitable characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.

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  1. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt–Universität Greifswald, Friedrich-Ludwig-Jahn–Str. 15a, 17487, Greifswald, Germany

    Armin Hemmerling

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Hemmerling, A. The Hausdorff-Ershov Hierarchy in Euclidean Spaces. Arch. Math. Logic 45, 323–350 (2006). https://doi.org/10.1007/s00153-005-0317-8

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  • DOI: https://doi.org/10.1007/s00153-005-0317-8

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