The high/low hierarchy in the local structure of the \(\omega\)-enumeration degrees (Q764275)

The paper is devoted to the local theory of \(\omega\)-enumeration degrees. It is proved that the local structure of the enumeration degrees is first-order definable as a substructure of the \(\omega\)-enumeration degrees. The definability of the classes \(H_{n}\) and \(L_{n}\) of the high\(_{n}\) and low\(_{n}\) \(\omega\)-enumeration degrees is shown. The interpretability of the first-order theory of true arithmetic in the local theory of the \(\omega\)-enumeration degrees is deduced.