Positive Presentations of Families in Relation to Reducibility with Respect to Enumerability
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The objects considered here serve both as generalizations of numberings studied in  and as particular versions of A-numberings, where 𝔸 is a suitable admissible set, introduced in  (in view of the existence of a transformation realizing the passage from e-degrees to admissible sets ). The key problem dealt with in the present paper is the existence of Friedberg (single-valued computable) and positive presentations of families. In , 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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