Geodesic
Negative numberings in admissible sets.~I
Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 47-92.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct an admissible set $\mathbb{A}$ such that the family of all $\mathbb{A}$-computably enumerable sets possesses a negative computable $\mathbb{A}$-numbering but lacks positive computable $\mathbb{A}$-numberings. We also discuss the question on existence of minimal negative $\mathbb{A}$-numberings. 
@article{MT_2023_26_1_a3,
     author = {I. Sh. Kalimullin and V. G. Puzarenko and M. Kh. Faizrahmanov},
     title = {Negative numberings in admissible {sets.~I}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {47--92},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/}
}
TY  - JOUR
AU  - I. Sh. Kalimullin
AU  - V. G. Puzarenko
AU  - M. Kh. Faizrahmanov
TI  - Negative numberings in admissible sets.~I
JO  - Matematičeskie trudy
PY  - 2023
SP  - 47
EP  - 92
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/
LA  - ru
ID  - MT_2023_26_1_a3
ER  - 
%0 Journal Article
%A I. Sh. Kalimullin
%A V. G. Puzarenko
%A M. Kh. Faizrahmanov
%T Negative numberings in admissible sets.~I
%J Matematičeskie trudy
%D 2023
%P 47-92
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/
%G ru
%F MT_2023_26_1_a3
I. Sh. Kalimullin; V. G. Puzarenko; M. Kh. Faizrahmanov. Negative numberings in admissible sets.~I. Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 47-92. http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/

[1] R. M. Friedberg, “Three theorems on recursive enumeration. I: Decomposition. II: Maximal set. III: Enumeration without duplication”, J. Symbolic Logic, 23:3 (1958), 309–316

[2] V. G. Puzarenko, “O razreshimykh vychislimykh $\mathbb{A}$-numeratsiyakh”, Algebra i logika, 41:5 (2002), 568–584

[3] V. G. Puzarenko, “K vychislimosti na spetsialnykh modelyakh”, Sib. mat. zhurn., 46:1 (2005), 185–208

[4] I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Fanzrakhmanov, “Pozitivnye numeratsii v dopustimykh mnozhestvakh”, Sib. mat. zhurn., 61:3 (2020), 607–621

[5] Yu. L. Ershov, Teoriya numeratsii, Nauka, M., 1977

[6] Yu. L. Ershov, Opredelimost i Vychislimost, Nauchnaya kniga (NI-IMIOO NGU), Novosibirsk, 1996

[7] J. Barwise, Admissible Sets and Structures: An Approach to Definability Theory, Springer-Verlag, Berlin–Heidelberg–New York, 1975

[8] Yu. L. Ershov, V. G. Puzarenko, A. I. Stukachev, “$\mathbb{HF}$-Computability”, COMPUTABILITY IN CONTEXT: Computation and Logic in the Real World, eds. S. B. Cooper, A. Sorbi, Imperial College Press, Singapore, 2011, 169–242

[9] R. I. Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin–Heidelberg–New York–London–Paris–Tokyo, 1987

[10] Yu. L. Ershov, Problemy razreshimosti i konstruktivnye modeli, Nauka, M., 1980

[11] S. S. Goncharov, Yu. L. Ershov, Konstruktivnye modeli, Nauchnaya kniga, Novosibirsk, 1999

[12] V. G. Puzarenko, “O vychislimosti nad modelyami razreshimykh teorii”, Algebra i logika, 39:2 (2000), 170–197

[13] B. F. Puzarenko, “Obobschennye numeratsii i opredelimost polya $\mathbb{R}$ v dopustimykh mnozhestvakh”, Vestnik NGU: ser. mat., mekh., inf., 2:3 (2003), 107–117

[14] A. S. Morozov, V. G. Puzarenko, “O $\Sigma$-podmnozhestvakh naturalnykh chisel”, Algebra i logika, 43:3 (2004), 291–320

[15] V. G. Puzarenko, I. Sh. Kalimullin, “O svodimosti na semeistvakh”, Algebra i logika, 48:1 (2009), 31–53

[16] J. C. Owings, “The meta r.e. sets but not the $\Pi_1^1$ sets can be enumerated without repetition”, J. Symbolic Logic, 35:2 (1970), 223–229

[17] W. Li, “Friedberg numbering in fragments of Peano arithmetics and $\alpha$-recursion theory”, J. Symbolic Logic, 78:4 (2013), 1135–1163

[18] I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Fanzrakhmanov, “Chastichnye razreshimye predstavleniya v giperarifmetike”, Sib. matem. zhurn., 60:3 (2019), 599–609

[19] I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Fanzrakhmanov, “O pozitivnykh i odnoznachnykh vychislimykh numeratsiyakh v giperarifmetike”, Algebra i logika, 59:1 (2020), 66–83

[20] M. Kh. Faizrahmanov, V. G. Puzarenko, “Absolute and relative properties of negatively-numbered families”, Lobachevskii J. Math., 42:4 (2021), 726–734