Direct and local denitions of the Turing jump

We show that there are 5 formulas in the language of the Turing degrees, D, with ,_ and ^, that de…ne the relations x00 y00, x00 = y00 and so x 2 L2(y) = fx yjx00 = y00g in any jump ideal containing 0(!). There are also 6& 6 and 8 formulas that de…ne the relations w = x00 and w = x0, respectively, in any such ideal I. In the language with just the quanti…er complexity of each of these de…nitions increases by one. For a lower bound on de…nability, we show that no 2 or 2 formula in the language with just de…nes L2 or L2(y). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is …xed on every degree above 000 and every relation on I which is invariant under the double jump or under join with 000 is de…nable over I if and only if it is de…nable in second order arithmetic with set quanti…cation ranging over sets whose degrees are in I.