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ON SUPERSETS OF NON-LOW$_2$ SETS

Part of: Computability and recursion theory

Published online by Cambridge University Press:  13 September 2021

KLAUS AMBOS-SPIES ,
ROD G. DOWNEY  and
MARTIN MONATH
KLAUS AMBOS-SPIES
Affiliation:
INSTITUT FÜR INFORMATIK UNIVERSITÄT HEIDELBERG IM NEUENHEIMER FELD 205 HEIDELBERGD-69120,GERMANYE-mail:ambos@math.uni-heidelberg.deE-mail:martin.monath@posteo.de
ROD G. DOWNEY
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITYWELLINGTON, P.O. BOX 600, NEW ZEALANDE-mail:rod.downey@vuw.ac.nz
MARTIN MONATH
Affiliation:
INSTITUT FÜR INFORMATIK UNIVERSITÄT HEIDELBERG IM NEUENHEIMER FELD 205 HEIDELBERGD-69120,GERMANYE-mail:ambos@math.uni-heidelberg.deE-mail:martin.monath@posteo.de
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Abstract

We solve a longstanding question of Soare by showing that if ${\mathbf d}$ is a non-low $_2$ computably enumerable degree then ${\mathbf d}$ contains a c.e. set with no r-maximal c.e. superset.

Keywords

computably enumerable setnon-low2r-maximal

MSC classification

Primary: 03D25: Recursively (computably) enumerable sets and degrees
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1282 - 1292
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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References

REFERENCES

Downey, R. and Shore, R., Degree theoretical definitions of the low 2 recursively enumerable sets, this Journal, vol. 60 (1995), pp. 727–756.Google Scholar
Lachlan, A., Degrees of recursively enumerable sets with no maximal supersets, this Journal, vol. 33 (1968), pp. 431–443.Google Scholar
Lerman, M., Degrees of Unsolvability , Springer, New York, 1983.10.1007/978-3-662-21755-9CrossRefGoogle Scholar
Martin, D., Classes of recursively enumerable sets and degrees of unsolvability. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.CrossRefGoogle Scholar
Shoenfield, J., Degrees of classes or r.e. sets, this Journal, vol. 41 (1976), pp. 695–696.Google Scholar
Soare, R.I., Automorphisms of the lattice of recursively enumerable sets, part II: Low sets. Annals of Mathematical Logic, vol. 22 (1982), pp. 69–107.CrossRefGoogle Scholar
Soare, R.I., Recursively Enumerable Sets and Degrees, Springer, New York, 1987.CrossRefGoogle Scholar