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

I. Sh Kalimullin, V. G. Puzarenko, M. Kh Faizrakhmanov

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1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)320-323
Number of pages4
JournalAlgebra and Logic
Issue number4
Publication statusPublished - 1 Sep 2018


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