A limit on relative genericity in the recursively enumerable sets

Journal of Symbolic Logic 54 (2):376-395 (1989)
  Copy   BIBTEX


Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X', its Turning jump, is recursive in $\varnothing'$ and high if X' computes $\varnothing''$ . 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 $A \bigoplus W$ 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 Turning 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 $(W \bigoplus A)'$ 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



    Upload a copy of this work     Papers currently archived: 92,261

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Complementation in the Turing degrees.Theodore A. Slaman & John R. Steel - 1989 - Journal of Symbolic Logic 54 (1):160-176.
Recursively enumerable generic sets.Wolfgang Maass - 1982 - Journal of Symbolic Logic 47 (4):809-823.
On recursive enumerability with finite repetitions.Stephan Wehner - 1999 - Journal of Symbolic Logic 64 (3):927-945.
An almost deep degree.Peter Cholak, Marcia Groszek & Theodore Slaman - 2001 - Journal of Symbolic Logic 66 (2):881-901.
On the orbits of hyperhypersimple sets.Wolfgang Maass - 1984 - Journal of Symbolic Logic 49 (1):51-62.
Recursive constructions in topological spaces.Iraj Kalantari & Allen Retzlaff - 1979 - Journal of Symbolic Logic 44 (4):609-625.


Added to PP

58 (#278,062)

6 months
18 (#144,337)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Infima in the d.r.e. degrees.D. Kaddah - 1993 - Annals of Pure and Applied Logic 62 (3):207-263.

Add more citations