Answering an open question raised by Cooper, we show that there exist Δ 2 0 sets D and E such that the singleton degree of E is a minimal cover of the singleton degree of D . This shows that the Σ 2 0 singleton degrees, and the Δ 2 0 singleton degrees, are not dense (and consequently the Π 2 0 Q -degrees, and the Δ 2 0 Q -degrees, are not dense). Moreover D and E can be built to lie in the same enumeration degree.

Abstract We show that there is a first order sentence φ(x: a, b, l) such that for every computable partial order and -degree u > 0e, there are -enumeration degrees a ≤ u, b, and l such that . Allowing to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the -enumeration degrees. Finally we show that there is a first order sentence that correctly identifies a subset of the standard models, which gives a parameterless interpretation of true arithmetic in the -enumeration degrees.

Abstract We show that every nonzero enumeration degree bounds a nonsplitting nonzero enumeration degree.

Abstract We show that in the language of { ≤ }. the Π3-fragment of the first order theory of the -enumeration degrees is undecidable. We then extend this result to show that the Π3-theory of any substructure of the enumeration degrees which contains the -degrees is undecidable.