TY - JOUR

T1 - Positive Presentations of Families in Relation to Reducibility with Respect to Enumerability

AU - Kalimullin, I. Sh

AU - Puzarenko, V. G.

AU - Faizrakhmanov, M. Kh

PY - 2018/9/1

Y1 - 2018/9/1

N2 - The objects considered here serve both as generalizations of numberings studied in [1] and as particular versions of A-numberings, where 픸 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.

AB - The objects considered here serve both as generalizations of numberings studied in [1] and as particular versions of A-numberings, where 픸 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.

UR - http://www.scopus.com/inward/record.url?scp=85056834967&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9503-8

DO - 10.1007/s10469-018-9503-8

M3 - Article

AN - SCOPUS:85056834967

VL - 57

SP - 320

EP - 323

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 4

ER -