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Nonhemimaximal degrees and the high/low hierarchy

Published online by Cambridge University Press:  12 March 2014

Fang Chengling  and
Wu Guohua
Fang Chengling
Affiliation:
Division of Mathematical Sciences, School of Physical andMathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore, E-mail: [email protected]:, [email protected]
Wu Guohua
Affiliation:
Division of Mathematical Sciences, School of Physical andMathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore, E-mail: [email protected]:, [email protected]
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Abstract

After showing the downwards density of nonhemimaximal degrees, Downey and Stob continued to prove that the existence of a low2, but not low, nonhemimaximal degree, and their proof uses the fact that incomplete m-topped degrees are low2 but not low. As commented in their paper, the construction of such a nonhemimaximal degree is actually a primitive 0‴ argument. In this paper, we give another construction of such degrees, which is a standard 0″-argument, much simpler than Downey and Stob's construction mentioned above.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 433 - 446
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Ambos-Spies, K., Jockusch, C.G. Jr., Shore, R.I., and Soare, R.I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109–128.CrossRefGoogle Scholar
[2] Downey, R. and Jockusch, C., T-degrees, jump classes and strong reducibilities, Transactions of the American Mathematical Society, vol. 301 (1987), pp. 103–136.CrossRefGoogle Scholar
[3] Downey, R., Lempp, S., and Shore, R., Highness and bounding minimal pairs. Mathematical Logic Quarterly, vol. 39 (1993), pp. 475–491.CrossRefGoogle Scholar
[4] Downey, R. and Stob, M., Jumps of hemimaximal sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 37 (1991), pp. 113–120.CrossRefGoogle Scholar
[5] Downey, R. and Stob, M., Automorphisms of the lattice of recursively enumerable sets: orbits, Advances in Mathematics, vol. 92 (1992), pp. 237–265.CrossRefGoogle Scholar
[6] Li, A., Wu, G., and Zhang, Z., A hierarchy for cuppable degrees, Illinois Journal of Mathematics, vol. 44 (2000), pp. 619–632.CrossRefGoogle Scholar
[7] Martin, D.A., Classes of recursively enumerable sets and degrees of unsolavability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.CrossRefGoogle Scholar
[8] Post, E.L., Recursively enumerable sets of positive integers and their decision problems, American Mathematical Society. Bulletin, vol. 50 (1944), pp. 284–316.CrossRefGoogle Scholar
[9] Soare, R.I., Automorphisms of the lattice of recursively enumerable sets. Part I: Maximal sets, Annals of Mathematics, vol. 100 (1974), pp. 80–120.CrossRefGoogle Scholar
[10] Soare, R.I., Recursively enumerable sets and degrees, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar