Let $T_1$ and $T_2$ be two first-order logical theories (over the same signature) such that $T_1 \subseteq T_2$ and both are recursively axiomatized.
My question is the following: is it possible that $T_1$ is finitely axiomatizable and decidable, while its extension $T_2$ is undecidable?
I would like to know whether there is or not a pair of logical theories with the previous requirements, but I have not been able to find any answer surfing the web. There is a paper by Verena Huber Dyson with the title "On the decision problem for theories of finite models" where there is an example for all these conditions except for requiring finitary axiomatizability. This paper is quite old (1964), so maybe someone here (if not, I will try mathoverflow) can provide me a better answer.
Addendum: By @JDH comments it is clear that
$T_1$ cannot be a complete theory (because then there are no consistent extensions).
$T_2$ cannot be finitely axiomatizable (because finite extensions of decidable theories are decidable)