Degree Structures

The Turing degrees 3d were introduced by Kleene and Post ([9], 1954) to isolate and study those properties of the subsets of the natural numbers N which are expressed purely in terms of relative computability. Intuitively, we form 3d by identifying any pair of subsets of N which are mutually computable and ordering the resulting equivalence classes by relative computability. The natural hierarchies of definability within arithmetic, analysis and higher fragments of set theory all have sharply focused images in 3d. We will focus our attention on the second order properties of the Turing degrees. We will phrase our discussion of the known results and especially of the techniques of their proofs within as large a context as possible, so to apply to degree structures based on other forms of relative definability, as well as to 3d, Formally, suppose that A and B are subsets of N, henceforth called reals, We say that A is Turing reducible to B (A T A, The Turing degrees D are the ==T-equivalence classes. The degree structure 2 associated with Turing reducibility is the partial ordering (D, <T), the Turing degrees with the ordering inherited from <T. 3d is the most intensively studied degree structure but not the only one. Although we cannot introduce them all in detail, we will point out a few other examples. Among the many degree structures on the real numbers obtained by varying the notion of relative definability, we might mention the arithmetic degrees, the hyperarithmetic degrees, the ^-degrees (A <Ek B if A is in the least set which is ^-admissible relative to B), the many-one degrees, the enumeration degrees, the ^-degrees and the degrees of constructibility. Alternately, we can retain relative computability and vary the class of reals on which it acts. We can form the Turing degrees of the recursively enumerable sets, the Turing degrees