There are $\Pi_5$ formulas in the language of the Turing degrees, D, with ≤, ∨ and $\vedge$ , that define the relations $x" \leq y"$ , x" = y" and so $x \in L_{2}(y)=\{x\geqy|x"=y"\}$ in any jump ideal containing $0^(\omega)$ . There are also $\Sigma_6$ & $\Pi_6$ and $\Pi_8$ formulas that define the relations w = x" and w = x', respectively, in any such ideal I. In the language with just ≤ the quantifier complexity of each of these definitions (...) increases by one. On the other hand, no Π2 or Σ2 formula in the language with just ≤ defines L2 or $x\inL_{2}(y)$ . 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 fixed on every degree above 0" and every relation on I that is invariant under double jump or joining with 0" is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in I. Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures. (shrink)

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated.