Archive for Mathematical Logic

, Volume 45, Issue 3, pp 323–350 | Cite as

The Hausdorff-Ershov Hierarchy in Euclidean Spaces

Article

Abstract

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta2 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 ΔB2 and Ershov's hierarchy in the class Δ02 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δta2. This is based on suitable characterizations of the sets from Δta2 which are obtained in a close analogy to those of the ΔB2 sets as well as those of the Δ02 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 ω2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.

Key words or phrases

Computable analysis Effective descriptive set theory Hausdorff's difference hierarchy Ershov's hierarchy Topological arithmetical hierarchy Resolvable sets Global and local depth of sets Recursivity in analysis Approximate decidability 

Mathematics Subject Classification (2000)

03D65 03D55 03E15 03F60 26E40 68Q01 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Mathematik und InformatikErnst-Moritz-Arndt–Universität GreifswaldGreifswaldGermany

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