Interpreting true arithmetic in the Δ02-enumeration degrees

Abstract

We show that there is a first order sentence φ(x; a, b, l) such that for every computable partial order 𝒫 and Δ⁰₂-degree u > 0_e, there are Δ⁰₂-enumeration degrees a ≤ u, b, and l such that 𝒫 ≅ {x : φ(x; a, b, l)}. Allowing 𝒫 to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the Δ⁰₂-enumeration degrees. Finally we show that there is a first order sentence that correctly identifies a subset of the standard models, which gives a parameterless interpretation of true arithmetic in the Δ⁰₂-enumeration degrees.