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An Application of 1-Genericity in the \(\Pi^0_2\) Enumeration Degrees

  • Liliana Badillo
  • Charles M. Harris
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)

Abstract

Using results from the local structure of the enumeration degrees we show the existence of prime ideals of Open image in new window enumeration degrees. We begin by showing that there exists a 1-generic enumeration degree Open image in new window which is noncuppable—and so properly downwards \(\Sigma^0_2\)—and low2. The notion of enumeration 1-genericity appropriate to positive reducibilities is introduced and a set A is defined to be symmetric enumeration 1-generic if both A and \(\ensuremath{\overline{A}} \) are enumeration 1-generic. We show that, if a set is 1-generic then it is symmetric enumeration 1-generic, and we prove that for any Open image in new window enumeration 1-generic set B the class \(\{\, X \,\mid \, \;\ensuremath{\negmedspace\leq_{\ensuremath{\mathrm{e}} }\negmedspace}\; B \,\}\) is uniform Open image in new window . Thus, picking 1-generic Open image in new window (from above) and defining Open image in new window it follows that every Open image in new window only contains Open image in new window sets. Since Open image in new window is properly \(\Sigma^0_2\) we deduce that Open image in new window contains no \(\Delta^0_2\) sets and so is itself properly Open image in new window .

Keywords

Induction Hypothesis Prime Ideal Minimal Pair Enumeration Degree Positive Reducibility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Liliana Badillo
    • 1
  • Charles M. Harris
    • 1
  1. 1.Department of MathematicsLeedsUK

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