Following Koellner in <http://plato.stanford.edu/entries/independence-large-cardinals/>, "a theory $T\_1$ is *interpretable* in $T\_2$ ($T\_1 \leq T\_2$) when, roughly speaking, there is a translation $\tau$ from the language of $T\_1$ to the language of $T\_2$ such that, for each sentence $\phi$ of the language of $T\_1$, if $T\_1\vdash \phi$ then $T\_2\vdash \tau (\phi)$."
Now take a theory $T$ and a statement $\phi$ independent from $T$, then for sure we have $T\leq T+\phi$ (with $\tau$ being trivial). Additionally, Koellner says, we can have either $T+\phi\not\leq T$ or $T+\phi\leq T$. But how is the second case possible, since the trivial proof $T+\phi\vdash\phi$ cannot be translated into a proof $T\vdash\phi$?


I know there are examples for the second case (like $T+\neg Con(T)\equiv T$, $T+CH\equiv T$, $T+\neg CH\equiv T$, where $\equiv$ means both $\leq$ and $\geq$), but how are they possible?



How can two theories $T$ and $T+\phi$ be mutually interpretable?
For example, $\newcommand\ZFC{\text{ZFC}}\ZFC+V=L$ is mutually interpretable with $\ZFC$ because inside any model of $\ZFC$ we may find its version of the constructible universe $L$, which is a model of $\ZFC+V=L$. So we have a definable way to interpret $\ZFC+V=L$ inside any model of $\ZFC$. The same argument works with $\ZFC+\text{GCH}\equiv\ZFC$.


Meanwhile, a theory like $\ZFC+\exists$ inaccessible cardinal is not mutually interpretable with $\ZFC$, because on consistency strength grounds we are not able uniformly to define a model of the former theory inside any model of the latter theory.


