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

Abstract

Using results from the local structure of the enumeration degrees we show the existence of prime ideals of enumeration degrees. We begin by showing that there exists a 1-generic enumeration degree 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 enumeration 1-generic set B the class $\{\, X \,\mid \, \;\ensuremath{\negmedspace\leq_{\ensuremath{\mathrm{e}} }\negmedspace}\; B \,\}$ is uniform . Thus, picking 1-generic (from above) and defining it follows that every only contains sets. Since is properly $\Sigma^0_2$ we deduce that contains no $\Delta^0_2$ sets and so is itself properly .