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Algebra and Logic

, Volume 57, Issue 4, pp 320–323 | Cite as

Positive Presentations of Families in Relation to Reducibility with Respect to Enumerability

  • I. Sh. KalimullinEmail author
  • V. G. Puzarenko
  • M. Kh. Faizrakhmanov
Article
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The objects considered here serve both as generalizations of numberings studied in [1] and as particular versions of A-numberings, where 𝔸 is a suitable admissible set, introduced in [2] (in view of the existence of a transformation realizing the passage from e-degrees to admissible sets [3]). The key problem dealt with in the present paper is the existence of Friedberg (single-valued computable) and positive presentations of families. In [3], it was stated that the above-mentioned transformation preserves the majority of properties treated in descriptive set theory. However, it is not hard to show that it also respects the positive (negative, decidable, single-valued) presentations. Note that we will have to extend the concept of a numbering and, in the general case, consider partial maps rather than total ones. The given effect arises under the passage from a hereditarily finite superstructure to natural numbers, since a computable function (in the sense of a hereditarily finite superstructure) realizing an enumeration of the hereditarily finite superstructure for nontotal sets is necessarily a partial function.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • I. Sh. Kalimullin
    • 1
    Email author
  • V. G. Puzarenko
    • 2
    • 3
  • M. Kh. Faizrakhmanov
    • 1
  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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