Interpreting N in the computably enumerable weak truth talble degrees

We give a rst-order coding without parameters of a copy of (N;+;) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter deenable subsets. Given a degree structure from computability theory, once the undecidability of its theory is known, an important further problem is the question of the actual complexity of the theory. If the structure is arithmetical, then its theory can be interpreted in true arithmetic, i.e. Th(N; +;). Thus an upper bound is ; (!) , the complexity of Th(N; +;). Here an interpretation of theories is a many{one reduction based on a computable map deened on sentences in some natural way. An example of an arithmetical structure is D T (; 0), the Turing{ degrees of 0 2 {sets. Shore 16] proved that true arithmetic can be interpreted in Th(D T (; 0)). A stronger result is interpretability without parameters of a copy of (N; +;) in the structure (interpretability of structures is deened in 8], Ch. 5). The main purpose of this paper is to prove such a result for the structure R wtt of computably enumerable weak truth table degrees. So far the undecidability of Th(R wtt) is known 3]. This result brings a program closer to its completion which has been carried out by various researchers over the past years: to determine the complexity of the theory for structures from computability theory. We discuss some results. For the c.e. many{one and Turing degrees, it has been proved that a copy of (N; +;) can be interpreted without parameters ((13] and 14], respectively). For the c.e. truth{table degrees and the lattice E of c.e. sets under inclusion, interpretations of Th(N; +;) in the theory have been given (for the rst, see 15]; the second result is due to Harrington, see 7]). In E one cannot interpret a copy of (N; +;) 7], which shows that the stronger, model theoretic result is not always implied by the mere interpretability of the theory of (N; +;). For the structures R m and R T of c.e. many{one and c.e. Turing degrees as well as for E, the methods employed (usually auxiliary codings of copies of (N; +;) with parameters and uniform deenability results) have been used to obtain further results of a model theoretic