Let $T_1$ and $T_2$ be two theories having the same set of symbols. Assume that any interpretation of $T_1$ is a model of $T_1$ if and only if it is not a model of $T_2$. Then:
$T_1$ and $T_2$ are finitely axiomatizable.
(i.e. there are finite sets of sentences $A_1$ and $A_2$ such that, for any sentence $S$: $T_1$ proves $S$ if and only if $A_1$ proves $S$, and $T_2$ proves $S$ if and only if $A_2$ proves $S$).
/The proof will be by contradiction; assume $T_1$ or $T_2$ are not finitely axiomatizable, then .....?/
Any one have any idea of how to prove this argument?