Geodesic
Turing degrees in refinements of the arithmetical hierarchy
Algebra i logika, Tome 57 (2018) no. 3, pp. 338-361.

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

We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to $\varnothing^n$, $n\omega$), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper.
Keywords: Ershov hierarchy, fine hierarchy, arithmetical hierarchy, Turing degrees. 
@article{AL_2018_57_3_a5,
     author = {V. L. Selivanov and M. M. Yamaleev},
     title = {Turing degrees in refinements of the arithmetical hierarchy},
     journal = {Algebra i logika},
     pages = {338--361},
     publisher = {mathdoc},
     volume = {57},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2018_57_3_a5/}
}
TY  - JOUR
AU  - V. L. Selivanov
AU  - M. M. Yamaleev
TI  - Turing degrees in refinements of the arithmetical hierarchy
JO  - Algebra i logika
PY  - 2018
SP  - 338
EP  - 361
VL  - 57
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2018_57_3_a5/
LA  - ru
ID  - AL_2018_57_3_a5
ER  - 
%0 Journal Article
%A V. L. Selivanov
%A M. M. Yamaleev
%T Turing degrees in refinements of the arithmetical hierarchy
%J Algebra i logika
%D 2018
%P 338-361
%V 57
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2018_57_3_a5/
%G ru
%F AL_2018_57_3_a5
V. L. Selivanov; M. M. Yamaleev. Turing degrees in refinements of the arithmetical hierarchy. Algebra i logika, Tome 57 (2018) no. 3, pp. 338-361. http://geodesic.mathdoc.fr/item/AL_2018_57_3_a5/

[1] S. B. Cooper, Degrees of unsolvability, Ph. D. thesis, Leicester Univ., England, 1971

[2] Yu. L. Ershov, “Ob odnoi ierarkhii mnozhestv. I”, Algebra i logika, 7:1 (1968), 47–74 | MR | Zbl

[3] C. G. Jockusch (jun.), R. A. Shore, “Pseudo-jump operators. II: Transfinite iterations, hierarchies and minimal covers”, J. Symb. Log., 49:4 (1984), 1205–1236 | DOI | MR | Zbl

[4] V. L. Selivanov, “Ob ierarkhii Ershova”, Sib. matem. zh., 26:1 (1985), 134–149 | MR | Zbl

[5] Yu. L. Ershov, “Ob odnoi ierarkhii mnozhestv. II”, Algebra i logika, 7:4 (1968), 15–47 | MR | Zbl

[6] Yu. L. Ershov, “Ob odnoi ierarkhii mnozhestv. III”, Algebra i logika, 9:1 (1970), 34–51 | MR

[7] V. L. Selivanov, “Ierarkhii giperarifmeticheskikh mnozhestv i funktsii”, Algebra i logika, 22:6 (1983), 666–692 | MR | Zbl

[8] V. L. Selivanov, “Fine hierarchies and Boolean terms”, J. Symb. Log., 60:1 (1995), 289–317 | DOI | MR | Zbl

[9] R. Rogers, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967 ; Kh. Rodzhers, Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR | Zbl

[10] V. L. Selivanov, “Tonkaya ierarkhiya arifmeticheskikh mnozhestv i opredelimye indeksnye mnozhestva”, Matematicheskaya logika i algoritmicheskie problemy, Tr. In-ta matem. SO AN SSSR, 12, 1989, 165–185 | Zbl

[11] W. Wadge, Reducibility and determinateness in the Baire space, Ph. D. thesis, Univ. California, Berkeley, 1984

[12] K. Kuratovskii, A. Mostovskii, Teoriya mnozhestv, Mir, M., 1970 | MR

[13] R. I. Soare, Recursively enumerable sets and degrees, Perspect. Math. Log. Omega Series, Springer-Verlag, Heidelberg a.o., 1987 ; R. I. Soar, Vychislimo perechislimye mnozhestva i stepeni, Kazanskoe matem. ob-vo, Kazan, 2000 | DOI | MR | Zbl