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On injective enumerability of recursively enumerable classes of cofinite sets

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Abstract

To date the problem of finding a general characterization of injective enumerability of recursively enumerable (r.e) classes of r.e. sets has proved intractable. This paper investigates the problem for r.e. classes of cofinite sets. We state a suitable criterion for r.e. classesC such that there is a boundn∈ω with |ω-A|≤n for allAC. On the other hand an example is constructed which shows that Lachlan's condition (F) does not imply injective enumerability for r.e. classes of cofinite sets. We also look at a certain embeddability property and show that it is equivalent with injective enumerability for certain classes of cofinite sets. At the end we present a reformulation of property (F).

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References

  • [K] Kummer, M.: Numberings ofR 1F, Computer Science Logic 1988. Lecture Notes in Computer Science385, 166–186 (1989)

    Google Scholar 

  • [KD] Kummer, M.: Beiträge zur Theorie der Numerierungen: Eindeutige Numerierungen. Dissertation, Universität Karlsruhe, Germany (1989)

    Google Scholar 

  • [L] Lachlan, A.: On recursive enumeration without repetition. Z. Math. Logik Grundlag. Math11, 209–220 (1965)

    Google Scholar 

  • [LM] Lachlan, A.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. (3)16, 537–569 (1966)

    Google Scholar 

  • [LC] Lachlan, A.: On recursive enumeration without repetition: A correction. Z. Math. Logik Grundlag. Math.13, 99–100 (1967)

    Google Scholar 

  • [M] Marchenkov, S.S.: On minimal numerations of systems of recursively enumerable sets (abstract). Sov. Math. Dokl.12, 843–845 (1971)

    Google Scholar 

  • [P] Pour-El, M.B., Howard, W.A.: A structural criterion for recursive enumeration without repetition. Z. Math. Logik Grundlag. Math.10, 105–114 (1964)

    Google Scholar 

  • [Yi] Yi, X.: Manuscript

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Thank you for technical support, Wolfgang Eppler, for intellectual support, Alistair Lachlan, and for proof-reading, Martin Kummer. Thanks also to the anonymous referee

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Wehner, S. On injective enumerability of recursively enumerable classes of cofinite sets. Arch Math Logic 34, 183–196 (1995). https://doi.org/10.1007/BF01375520

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  • DOI: https://doi.org/10.1007/BF01375520

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