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

Abstract

There are Π₅ formulas in the language of the Turing degrees, 𝒟, with ≤, ∨ and ∧, that define the relations x''≤ y'', x''=y'' and so x∈ L₂( y)={x≥ y| x''=y''} in any jump ideal containing 0^(ω). There are also Σ₆ & Π₆ and Π₈ 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∈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 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.