Let $T$ be a recursively axiomatizable consistent theory formulated in classical first-order logic and assume $\mathsf{ZFC}$ is consistent.
Does there always exist a recursively axiomatizable consistent extension of $\mathsf{ZFC}$ in which T is interpretable?
It is well known that almost all of recursively axiomatizable consistent first-order theories can be interpreted in $\mathsf{ZFC}$ or its recursively axiomatizable consistent extension.
Some first-order theories such as Presburger arithmetic or $\mathsf{RCF}$ are complete and then there is no consistent essential extension of them, on the other hand some first-order theories satisfying some conditions such a $\mathsf{ZFC}$ have infinitely many recursively axiomatizable consistent essential extensions according to Gödel's first incompleteness theorem.
Then, I wonder if the proof theoretic powers of all of the recursively axiomatizable consistent extensions of $\mathsf{ZFC}$ are bounded or not in the class of all recursively axiomatizable consistent first-order theories.
In more general, some theories in other logical system such as higher-order logic or type system can be, in a sense, interpreted in $\mathsf{ZFC}$ or its recursively axiomatizable consistent extension. But here I restrict to first-order theories for the notion of interpretability to be clear.
Any help would be appreciated.
I think Asaf's comment resolves my question.
Any first-order theory that be able to prove completeness theorem, and define provability and then consistency can build a model of any recursively axiomatizable first-order theory $\mathsf{T}$ therefore interpret $\mathsf{T}$ with the added assumption $\mathsf{Con(T)}$.
I should have resolved this by myself. But, my understanding about the notion of interpretability seems insufficient in a basic level.
Thank you for all comennts!