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On Non-principal Arithmetical Numberings and Families

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Abstract

The paper studies \(\varvec{\Sigma ^0_n}\)-computable families (\(\varvec{n\geqslant 2}\)) and their numberings. It is proved that any non-trivial \(\varvec{\Sigma ^0_n}\)-computable family has a complete with respect to any of its elements \(\varvec{\Sigma ^0_n}\)-computable non-principal numbering. It is established that if a \(\varvec{\Sigma ^0_n}\)-computable family is not principal, then any of its \(\varvec{\Sigma ^0_n}\)-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal \(\varvec{\Sigma ^0_n}\)-computable numberings. It is also shown that for any \(\varvec{\Sigma ^0_n}\)-computable numbering \(\varvec{\nu }\) of a \(\varvec{\Sigma ^0_n}\)-computable non-principal family there exists its \(\varvec{\Sigma ^0_n}\)-computable numbering that is incomparable with \(\varvec{\nu }\). If a non-trivial \(\varvec{\Sigma ^0_n}\)-computable family contains the least and greatest elements under inclusion, then for any of its \(\varvec{\Sigma ^0_n}\)-computable non-principal non-least numberings \(\varvec{\nu }\) there exists a \(\varvec{\Sigma ^0_n}\)-computable numbering of the family incomparable with \(\varvec{\nu }\). In particular, this is true for the family of all \(\varvec{\Sigma ^0_n}\)-sets and for the families consisting of two inclusion-comparable \(\varvec{\Sigma ^0_n}\)-sets (semilattices of the \(\varvec{\Sigma ^0_n}\)-computable numberings of such families are isomorphic to the semilattice of \(\varvec{m}\)-degrees of \(\varvec{\Sigma ^0_n}\)-sets).

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Acknowledgements

This work was supported by the Russian Science Foundation (grant no. 23-21-00181) and performed under the development programme of the Volga Region Mathematical Center (agreement no. 075-02-2023-944).

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  1. Volga Region Mathematical Center, Kazan Federal University, 35 Kremlevskaya Str., Kazan, 420008, Tatarstan, Russia

    Marat Faizrahmanov

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  1. Marat Faizrahmanov
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M.F. obtained all the results and wrote the manuscript alone. All the results are new and have not been published anywhere before.

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Correspondence to Marat Faizrahmanov.

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Faizrahmanov, M. On Non-principal Arithmetical Numberings and Families. Theory Comput Syst 68, 271–282 (2024). https://doi.org/10.1007/s00224-024-10165-z

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