A degree-theoretic definition of the ramified analytical hierarchy

AbstractLet D be the set of all (Turing) degrees, < the usual partial ordering of D and j the (Turing) jump operator on D. The following relations are shown to be first-order definable in the structure D = 〈D, ⩽, j〉 : d1 is hyperarithmetical in d2, d1 is the hyperjump of d2, d1 is ramified analytical in d2 (Corollaries 4.6,4.13,4.16). A first-order, degree theoretic definition of the ramified analytical hierarchy is obtained (Theorem 5.6). A first-order sentence is found which is true in D if the universe is (a generic extension of) L, and false in D if 0# exists (Corollary 4.7). The question of whether the notion of uniform upper bound is degree theoretically definable is investigated (Section 6). Exact pairs of upper bounds are used to replace analytical definability by arithmetical definability (Theorem 3.1)