We point out an existence criterion for positive computable total \( {\Pi}_1^1 \) -numberings of families of subsets of a given \( {\Pi}_1^1 \) -set. In particular, it is stated that the family of all \( {\Pi}_1^1 \) -sets has no positive computable total \( {\Pi}_1^1 \) -numberings. Also we obtain a criterion of existence for computable Friedberg \( {\Sigma}_1^1 \) -numberings of families of subsets of a given \( {\Sigma}_1^1 \) - set, the consequence of which is the absence of a computable Friedberg \( {\Sigma}_1^1 \) -numbering of the family of all \( {\Sigma}_1^1 \) -sets. Questions concerning the existence of negative computable \( {\Pi}_1^1 \) - and \( {\Sigma}_1^1 \) -numberings of the families mentioned are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. C. Owings, Jr., “The meta-r.e. sets, but not the \( {\Pi}_1^1 \) sets, can be enumerated without repetition,” J. Symb. Log., 35, No. 2, 223-229 (1970).
I. Sh. Kalimullin, V. G. Puzarenko, and M. Kh. Faizrakhmanov, “Partial decidable presentations in hyperarithmetic,” Sib. Math. J., 60, No. 3, 464-471 (2019).
I. Sh. Kalimullin, V. G. Puzarenko, and M. Kh. Faizrakhmanov, “Positive presentations of families in relation to reducibility with respect to enumerability,” Algebra and Logic, 57, No. 4, 320-323 (2018).
J. Barwise, Admissible Sets and Structures. An Approach to Definability Theory, Perspect. Math. Log., Springer, Berlin (1975).
H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).
G. E. Sacks, Higher Recursion Theory, Perspect. Math. Log., Springer, Berlin (1990).
M. V. Dorzhieva, “Elimination of metarecursive in Owing’s theorem,” Vestnik NGU, Mat., Mekh., Inf., 14, No. 1, 35-43 (2014).
M. V. Dorzhieva, “Undecidability of elementary theory of Rogers semilattices in analytical hierarchy,” Sib. El. Mat. Izv., 13, 148-153 (2016); http://semr.math.nsc.ru/v13/p148-153.pdf.
M. V. Dorzhieva, “Friedberg numbering of the family of all \( {\Sigma}_2^1 \) -sets,” Sib. Zh. Ch. Prikl. Mat., 18, No. 2, 47-52 (2018).
V. G. Puzarenko, “Decidable computable 𝔸-numberings,” Algebra and Logic, 41, No. 5, 314- 322 (2002).
Yu. L. Ershov, Definability and Computability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk, Ekonomika, Moscow (2000).
Yu. L. Ershov, Numeration Theory [in Russian], Nauka, Moscow (1977).
I. Sh. Kalimullin and V. G. Puzarenko, “Computability principles on admissible sets,” Mat. Trudy, 7, No. 2, 35-71 (2004).
I. Sh. Kalimullin, V. G. Puzarenko, and M. Kh. Faizrakhmanov, “Positive presentations of families relative to e-oracles,” Sib. Math. J., 59, No. 4, 648-656 (2018).
S. S. Goncharov and A. Sorbi, “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, 36, No. 6, 359-369 (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by RFBR (project No. 18-01-00574) and by the Russian Ministry of Education and Science (project No. 1.451.2016/1.4).
Supported by RFBR (project No. 18-01-00624) and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2019-0003).
Supported by Russian Science Foundation, project No. 18-11-00028.
Translated from Algebra i Logika, Vol. 59, No. 1, pp. 66-83, January-February, 2020.
Rights and permissions
About this article
Cite this article
Kalimullin, I.S., Puzarenko, V.G. & Faizrakhmanov, M.K. Computable Positive and Friedberg Numberings in Hyperarithmetic. Algebra Logic 59, 46–58 (2020). https://doi.org/10.1007/s10469-020-09578-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-020-09578-9