Can there exist a consistent, recursively axiomatizable theory $T$, such that $\forall \phi, TA\vdash \phi \Rightarrow$ $T\vdash \tau(\phi)$, where $\tau$ is some suitable translation from the language of $TA$ to that of $T$?

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**Edit:** By "suitable translation" I was thinking about a translation that preserves the intended meaning of $\phi$ in $TA$. For example, for any theorem $\phi$ of $PA$, you can also prove it ($\tau(\phi)$) in the language of $ZFC$, and $\phi$ and $\tau(\phi)$ share the same intended meaning in the standard model $\mathcal{N}$ of arithmetic. Only now instead of $PA$ and $ZFC$, we're thinking about $TA$ and any recursively axiomatizable theory.

 (I apologize if this sounds too sloppy. I had some trouble getting this part straight in my head, but I hope the point is understood)