## Interpreting N in the computably enumerable weak truth table degrees

### BibTeX

@MISC{Nies_interpretingn,

author = {André Nies},

title = {Interpreting N in the computably enumerable weak truth table degrees},

year = {}

}

### OpenURL

### Abstract

We give a first-order coding without parameters of a copy of (N;+; \Theta) in the computably enumerable weak truth table degrees. As a tool,we develop a theory of parameter definable 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; +; \Theta). Thus an upper bound is ; (!) , the complexity of Th(N; +; \Theta). Here an interpretation of theories is a many--one reduction based on a computable map defined on sentences in some natural way. An example of an arithmetical structure is D T ( ; 0 ), the Turing-- degrees of \Delta 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; +; \Theta) in the structure (interpretability of struc...