We investigate definability in $\mathcal{R}$, therecursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. Firstwe show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove thatthe recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which impliesthat 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 arithmeticand invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare'sQuestion 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), ChapterXVI).
1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.
