Just a question about interpretations which I'm not sure of:
Say we have two theories $T_0$ and $T_1$. Then an interpretation $I$ of $T_0$ into $T_1$ is an interpretation $I$ of the language $L_0$ of $T_0$ into $T_1$ s.t. for every $L_0$-sentence $\phi$, $\phi\in T_0\implies \phi^I\in T_1$
Now if $T_0$ is axiomatizable by some $\Gamma$, doesn't it suffice to have that $\phi \in \Gamma \implies \phi ^I\in T_1$ for every $L_0$-sentence $\phi$?
Any help is appreciated
-Thanks
I am not sure what definitions you are working with, but I will suppose that $L_0$ and $L_1$ are two different first order languages without constants or function symbols and that an interpretation $I$ is given by a formula $D(x)$ of $L_1$ giving the domain plus formulas of $L_1$ for each relation relation symbol of $L_0$. I think sometimes people use more generous notions of interpretation, but it probably doesn't matter.
In answer to the first question, yes, you just have to show that interpretation preserves logical consequence.
In answer to the second, do you mean $\{\phi^I : \phi\in T_0\}$? That's not true in general because formulas of the form $\phi^I$ will typically be very special.
