Definability in the enumeration degrees

Abstract

We prove that every countable relation on the enumeration degrees, ${\frak E}$ , is uniformly definable from parameters in ${\frak E}$ . Consequently, the first order theory of ${\frak E}$ is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order theory of the enumeration degrees of the $\Sigma^0_2$ sets is not decidable.