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Elementary theories and hereditary undecidability for semilattices of numberings

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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.

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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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Correspondence to Manat Mustafa.

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This work is supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. N. Bazhenov was partially supported by RFBR, according to the research Project No. 16-31-60058 mol_a_dk. M. Yamaleev was partially supported by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, Project No. 1.1515.2017/4.6.

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Bazhenov, N., Mustafa, M. & Yamaleev, M. Elementary theories and hereditary undecidability for semilattices of numberings. Arch. Math. Logic 58, 485–500 (2019). https://doi.org/10.1007/s00153-018-0647-y

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