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Fine hierarchy and definable index sets. (English. Russian original) Zbl 0780.03019

Algebra Logic 30, No. 6, 463-475 (1991); translation from Algebra Logika 30, No. 6, 705-725 (1991).
Summary: This paper, a sequel to two earlier papers [the author, Tr. Inst. Mat. 12, 165-185 (1989; Zbl 0746.03038); Algebra Logika 29, No. 2, 220-240 (1990)], is devoted to the problem of characterizing the \(m\)-degrees of definable index sets. In section 1 we determine fairly general conditions which are sufficient for universality in the classes of the fine hierarchy of the first quoted paper. These conditions are then used to obtain results about index sets in numberings of recursively enumerable (r.e.) Boolean algebras, sentences, r.e. sets and r.e. \(m\)-degrees.

MSC:

03D55 Hierarchies of computability and definability
03D25 Recursively (computably) enumerable sets and degrees
03D45 Theory of numerations, effectively presented structures

Citations:

Zbl 0746.03038
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Full Text: DOI

References:

[1] V. L. Selivanov, ”Fine hierarchies of arithmetical sets and definable index sets,” Tr. Inst. Mat. Sib. Otd. Akad. Nauk SSSR,12, 165–185 (1989). · Zbl 0746.03038
[2] V. L. Selivanov, ”Index sets of classes of hyperhypersimple sets,” Algebra Logika,29, No. 2, 220–240 (1990).
[3] V. L. Selivanov, ”Fine hierarchies of formulas,” Algebra Logika,30, No. 5, 568–582 (1991).
[4] S. P. Odintsov and V. L. Selivanov, ”Arithmetical hierarchy and ideals of numbered Boolean algebras,” Sib. Mat. Zh.,30, No. 6, 140–149 (1989). · Zbl 0711.03016
[5] W. Hanf, ”The Boolean algebra of logic,” Bull. Am. Math. Soc.,31, No. 3, 587–589 (1975). · Zbl 0324.02044 · doi:10.1090/S0002-9904-1975-13747-5
[6] M. G. Peretyat’kin, ”Finitely axiomatizable totally transcendental theories,” Tr. Inst. Mat. Sib. Otd. Akad. Nauk SSSR,2, 88–134 (1982).
[7] V. L. Selivanov, ”Hierarchies of hyperarithmetical sets and functions,” Algebra Logika,22, No. 6, 666–692 (1983). · Zbl 0536.03025
[8] V. L. Selivanov, ”Index sets of factor objects of the Post numbering,” Algebra Logika,27, No. 3, 343–358 (1988).
[9] V. L. Selivanov, ”Application of precomplete numberings to degress of tabular type and index sets,” Algebra Logika,28, No. 1, 75–82 (1989). · Zbl 0687.03024
[10] A. Lachlan, ”Recursively enumerable many-one degrees,” Algebra Logika,11, No. 3, 326–358 (1972). · Zbl 0282.02015 · doi:10.1007/BF02330746
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