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Some contrasts between degrees and the arithmetical hierarchy

Published online by Cambridge University Press:  12 March 2014

Alfred B. Manaster
Alfred B. Manaster*
Affiliation:
University of California, San Diego
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Extract

The nonrecursiveness of an arithmetical set can be measured in at least two ways. One way is to consider the level of the set in the arithmetical hierarchy. Another is to consider the degree of unsolvability of the set; more precisely to look at the set of degrees recursive in the degree of the set. The Hierarchy Theorem [1] indicates a correspondence between these two ways of considering the nonrecursiveness of an arithmetical set. The results below indicate some of the contrasts between these two points of view. (It should perhaps be mentioned that, although the results are stated in terms of degrees and levels in the arithmetical hierarchy, they can also be stated, in view of the Hierarchy Theorem, strictly in terms of degrees.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 301 - 304
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[2]Sacks, G. E., A minimal degree less than 0′, Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416–419.CrossRefGoogle Scholar
[3]Sacks, G. E., Degrees of unsolvability. Annals of Mathematics Studies No. 55, Princeton University Press, 1963.Google Scholar
[4]Sasso, L. P. Jr., A cornucopia of minimal degrees, this Journal, vol. 35 (1970), pp. 383–388.Google Scholar
[5]Spector, C., On degrees of recursive unsolvability, Annals of mathematics, vol. 64 (1966), pp. 581–592.CrossRefGoogle Scholar