Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-01T03:06:09.882Z Has data issue: false hasContentIssue false
  • English
  • Français

A limit on relative genericity in the recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Steffen Lempp  and
Theodore A. Slaman
Steffen Lempp*
Affiliation:
Department of Mathematics, Yale University, New Haven, Connecticut 06520
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
*
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Get access Rights & Permissions [Opens in a new window]

Abstract

Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.

Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.

We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 376 - 395
Copyright
Copyright © Association for Symbolic Logic 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BMta]Bickford, M. and Mills, C. F., Lowness properties of r.e. sets (to appear).Google Scholar
[Jo85]Jockusch, C. G., Genericity for recursively enumerable sets, Recursion theory week, Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, Berlin, 1985, pp. 203–232.CrossRefGoogle Scholar
[La66]Lachlan, A. H., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, ser. 3, vol. 16 (1966), pp. 537–569.CrossRefGoogle Scholar
[Ma66]Martin, D. A., 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
[Ms82]Maass, W., Recursively enumerable generic sets, this Journal, vol. 47 (1982), pp. 809–823.Google Scholar
[Sa63a]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, ser. 2, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[Sa63b]Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223–239.CrossRefGoogle Scholar
[Sa64]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, ser. 2, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[Shta]Shore, R. A., A non-inversion theorem for the jump operator, Annals of Pure and Applied Logic (to appear).Google Scholar
[So85]Soare, R. I., Tree arguments in recursion theory and the 0‴ priority method, Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 53–106.CrossRefGoogle Scholar
[So87]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[SS82]Soare, R. I. and Stob, M., Relative recursive enumerability, Proceedings of the Herbrand symposium, North-Holland, Amsterdam, 1982, pp. 299–324.CrossRefGoogle Scholar