Interpretability and Definability in the Recursively Enumerable Degrees

Abstract

We investigate definability in 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 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 Lown and Highn−1 n ⩾ 2 are definable in R without parameters and, more generally, that all relations on R that are definable in arithmetic and invariant under the double jump are actually definable in R. This partially answers Soare's Question 3.7 (R. Soare, Recursively enumerable sets and degrees (Springer, Berlin, 1987), Chapter XVI). 1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.