Interpretability and definability in therecursively enumerable degrees

Abstract

We investigate definability in $\mathcal{R}$, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove that the recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes $\mathrm{Low}_n$ and $\mathrm{High}_{n-1}$$(n\ge 2)$ are definable in $\mathcal{R}$ without parameters and, more generally, that all relations on $\mathcal{R}$ that are definable in arithmetic and invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare's Question 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), Chapter XVI).

1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.