By not interpreting arithmetic, I mean it does not interprets enough arithmetic for Godel's argument (coding the syntax, finding the fix point) to work through. In other words, is there any other methods to prove that a theory does not have a computable consistent complete extension, or can we prove the converse that every such theory interprets arithmetic?


Related, is there a class of structures whose first-order theory is not computable and arithmetic is not interpreted in it?



Is there a first-order theory who does not interpret arithmetic yet still does not have a computable consistent completion?
Any theory that can represent all recursive functions has no consistent decidable extension, however there are such theories that do not interpret even as weak an arithmetic as Robinson’s theory $R$, see my paper *Recursive functions and existentially closed structures* ([arXiv:1710.09864](https://arxiv.org/abs/1710.09864) [math.LO], to appear in Journal of Mathematical Logic).


You don’t even need to represent all recursive functions. Fix a recursively inseparable pair of r.e. predicates $A,B\subseteq\mathbb N$. Let $L$ be the language consisting of one unary predicate $P$, and constants $\overline n$ for every $n\in\mathbb N$, and let $T$ be the (recursively axiomatizable) theory axiomatized by $P(\overline n)$ for $n\in A$, and $\neg P(\overline n)$ for $n\in B$. Then $T$ has no decidable consistent extension, and it does not interpret much of anything (in particular, if $S$ is a theory in a *finite* language interpretable in $T$, then $S$ is also interpretable in a finite-language fragment of $T$, and as such does have a decidable extension).


