# Is incompleteness preserved by interpretations?

Let $$T$$ be an incomplete theory, in the sense that there is a formula $$\phi$$ of the language of $$T$$ and there are two models of $$T$$, $$M$$ and $$M'$$, such that $$\phi$$ is true in $$M$$ and $$\neg \phi$$ is true in $$M'$$. Suppose that $$T$$ is interpretable in a theory $$T'$$, in the sense that there is a translation from the language of $$T$$ to the language of $$T'$$ which preserves logical form and such that theorems of $$T$$ are mapped to theorems of $$T'$$.

Does it follow, in general, that $$T'$$ is incomplete?

No. For example, let $$T'$$ be your favorite complete theory, and let $$T$$ be the empty theory in the same language (with no axioms). $$T$$ is incomplete.
But there is a trivial interpretation of $$T$$ in $$T'$$: take the identity translation from the language of $$T$$ to the language of $$T'$$. Every theorem of $$T$$ (which are just the first-order validities) translates to a theorem of $$T'$$.
Maybe I should add that there is a notion of an "essentially undecidable theory". This is a theory $$T$$ such that for any theory $$T'$$ interpreting $$T$$, the set of consequences of $$T'$$ is undecidable.
In particular, $$T$$ is incomplete, and if $$T'$$ is a computably enumerable theory interpreting $$T$$, then $$T'$$ is incomplete (since the set of consequences of a complete computably enumerable theory is decidable).