On the Symmetric Enumeration Degrees

A set A is symmetric enumeration (se-) reducible to a set B (A≤seB) if A is enumeration reducible to B and A is enumeration reducible to B. This reducibility gives rise to a degree structure (Dse) whose least element is the class of computable sets. We give a classification of ≤se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (DT) into the enumeration degrees (De) translates to an embedding ( ιse ) into Dse that preserves least element, suprema and infima. We define a weak and a strong jump and we observe that ιse preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers and diamond embeddings in Dse. We show that certain classes of se-degrees are first order definable, in particular the classes of semirecursive, Σn ∪Πn, ∆n (for any n ∈ ω), and embedded Turing degrees. This last result allows us to conclude that the theory of Dse has the same 1-degree as the theory of Second Order Arithmetic. Date: September 24, 2006.