Local denitions in degree structures: the Turing jump, hyperdegrees and beyond

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. On the other hand, no 2 or 2 formula in the language with just de…nes L2 or x 2 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 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. Similar direct coding arguments show that that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quanti…cation ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures.