Degree Theoretic Definitions Of The Low 2 Recursively Enumerable Sets

this paper. Clearly the most remarkable result relating the jump operator to the ordering of degrees is Cooper's recent theorem [1993] that the jump operator is definable, from ordering alone, in DT , the Turing degrees with T . Slaman and Woodin [1996] have used this result to considerably improve our knowledge about definability within, and possible automorphisms of, DT . One of their most remarkable results connects DT with RT , the Turing degrees of the r. e. sets: If RT is rigid (i.e., has no nontrivial automorphisms), then so is DT . Although much is known about the global structure of the degrees as a whole under many reducibilities (see for example Nerode and Shore [1980 a,b] and Shore [1985] as well as Slaman and Woodin [1996]), almost nothing in this vein is known about the r. e. degrees. We believe that, as in the degrees as a whole, a key problem for the study of the r. e. sets and degrees should be the elucidation of the relation between their degree structure and the jump operator. The standard approach to this problem is to examine the properties of the jump classes