TITLE: Is there a first-order theory who does not interpret arithmetic yet still does not have a computable consistent completion?
QUESTION [17 upvotes]: 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?

REPLY [6 votes]: Another example is the theory $\text{Th}(\mathcal{P}(\mathbb{R}),\subset,<)$ of subsets of the reals, where $S$ is less than $S'$ iff every element of $S$ is less than every element of $S'$.
The theory is consistent and complete by definition. Shelah proved that it is not decidable, which means it also has no decidable extension, and Gurevich and Shelah jointly proved that it does not interpret arithmetic.
To be more precise, Gurevich and Shelah work with a variant of the theory where $<$ applies only to singletons, and they prove that it does not interpret even the weak set theory of null set, singleton and union:
\begin{align}
\exists y \forall z &[z \notin y]\\
\forall x \exists y \forall z &[z \in y\leftrightarrow z=x]\\
\forall w \forall x \exists y \forall z &[z \in y\leftrightarrow z \in w \text{ or } z \in x]\\
\end{align}
The two results on undecidability and non-interpretability are both difficult, but these questions of mine have more details.