Enumeration Degrees Are Not Dense (1996)
BibTeX
@MISC{Calhoun96enumerationdegrees,
author = {William Calhoun and Theodore A. Slaman},
title = {Enumeration Degrees Are Not Dense},
year = {1996}
}
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Abstract
We show that the \Pi 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper [3] showed that the enumeration degrees (e-degrees) are not dense. He also showed that the e-degrees of the \Sigma 0 2 sets are dense [2]. Since Cooper's nondensity proof constructs a \Sigma 0 7 set, he posed the question: What is the least n (2 n 6) such that the e-degrees below 0 n are not dense? (We consider the Turing degrees to be a subset of the e-degrees by identifying the Turing degree of a set A with the e-degree of A \Phi A.) Cooper also conjectured that the e-degrees of the \Pi 0 2 sets are dense. Here we show that the e-degrees are not dense in the \Pi 0 2 e-degrees. Thus the answer to Cooper's question is 2 and his conjecture is false. This means that Cooper's proof that the \Sigma 0 2 e-degrees are dense cannot be extended beyond the \Sigma 0 2 level in the arithmetical hierarchy. 2 Description of the Construction The basic plan of the cons...
