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Archive for Mathematical Logic

, Volume 58, Issue 3–4, pp 485–500 | Cite as

Elementary theories and hereditary undecidability for semilattices of numberings

  • Nikolay Bazhenov
  • Manat MustafaEmail author
  • Mars Yamaleev
Article

Abstract

A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their first order theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. In this paper, we study decidability for theories of upper semilattices that arise from the theory of numberings. We use the following approach: given a level of complexity, say \(\varSigma ^0_{\alpha }\), we consider the upper semilattice \(R_{\varSigma ^0_{\alpha }}\) of all \(\varSigma ^0_{\alpha }\)-computable numberings of all \(\varSigma ^0_{\alpha }\)-computable families of subsets of \(\mathbb {N}\). We prove that the theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show that the theory of the semilattice of all numberings is computably isomorphic to second order arithmetic. We also obtain a lower bound for the 1-degree of the theory of the semilattice of all \(\varSigma ^0_{\alpha }\)-computable numberings, where \(\alpha \ge 2\) is a computable successor ordinal. Furthermore, it is shown that for any of the theories T mentioned above, the \(\varPi _5\)-fragment of T is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on \(\mathbb {N}\), equipped with composition.

Keywords

Computability theory Numbering theory Computably enumerable equivalence relation Hereditary undecidability Elementary definability First order arithmetic Second order arithmetic Upper semilattice Rogers semilattice 

Mathematics Subject Classification

03D45 

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Notes

Acknowledgements

Part of the research contained in this paper was carried out while the first and the last authors were visiting the Department of Mathematics of Nazarbayev University, Astana. The authors wish to thank Nazarbayev University for its hospitality. The authors also thank the anonymous reviewers for their helpful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Department of Mathematics, School of Science and TechnologyNazarbayev UniversityAstanaKazakhstan
  4. 4.Kazan Federal UniversityKazanRussia

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