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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References
Yu. L. Ershov, Numeration Theory [in Russian], Nauka, Moscow (1977).
Yu. L. Ershov, Definability and Computability, Sib. School Alg. Log. [in Russian], 2nd ed., Ekonomika, Moscow (2000).
I. Sh. Kalimullin and V. G. Puzarenko, “Computability principles on admissible sets,” Mat. Trudy, 7, No. 2, 35-71 (2004).
R. I. Soare, Recursively Enumerable Sets and Degrees. A Study of Computable Functions and Computably Generated Sets, Persp. Math. Log., Omega Ser., Springer, Berlin (1987).
S. S. Goncharov and A. Sorbi, “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, 36, No. 6, 359-369 (1997).
S. A. Badaev, “Positive enumerations,” Sib. Math. J., 18, No. 3, 343-352 (1977).
S. S. Goncharov, S. Lempp, and D. R. Solomon, “Friedberg numberings of families of ncomputably enumerable sets,” Algebra and Logic, 41, No. 2, 81-86 (2002).
G. E. Sacks, Higher Recursion Theory, Springer, Berlin (1990).
J. C. Owings, Jun., “The meta-r.e. sets, but not the Π1 1 sets, can be enumerated without repetition,” J. Symb. Log., 35, No. 2, 223-229 (1970).
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∗Supported by Russian Science Foundation, project No. 18-11-00028, and by the Russian Ministry of Education and Science.
∗∗Supported by RFBR, project No. 18-01-00624-a.
∗∗∗Supported by the Russian Ministry of Education and Science, project No. 1.1515.2017/4.6.
Translated from Algebra i Logika, Vol. 57, No. 4, pp. 492-498, July-August, 2018.
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Kalimullin, I.S., Puzarenko, V.G. & Faizrakhmanov, M.K. Positive Presentations of Families in Relation to Reducibility with Respect to Enumerability. Algebra Logic 57, 320–323 (2018). https://doi.org/10.1007/s10469-018-9503-8
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DOI: https://doi.org/10.1007/s10469-018-9503-8