Abstract

We show that there is a first order sentence φ(x; a, b, l) such that for every computable partial order $\scr{P}$ and $\Delta _{2}^{0}\text{-degree}$ u > 0 e , there are $\Delta _{2}^{0}\text{-enumeration}$ degrees a ≤ u, b, and l such that $\scr{P}\cong \{\mathbf{\mathit{x}}\colon \varphi (\mathbf{\mathit{x}};\mathbf{\mathit{a}},\mathbf{\mathit{b}},\mathbf{\mathit{l}})\}$ . Allowing $\scr{P}$ to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the $\Delta _{2}^{0}\text{-enumeration}$ 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 $\Delta _{2}^{0}\text{-enumeration}$ degrees