We have a theory in first-order logic which we know that is uncountably categorical, complete but not finitely axiomatisable. We also want to know if it is decidable. But I don't know the procedure for finding this out. Can someone give me a starting point?
I have the theory given explicitly but I want to avoid sharing too much here since it is a homework assignment and I need to solve it by myself.
Edit: According to comments, we have that a theory is decidable if it is recursively axiomatized. Can someone show me why or direct me to the theorem?
If the theory is recursively axiomatizable by a set of axioms $T$, then the predicate "$n$ encodes a proof from $T$" is recursive.
Since $T$ is complete, for every $\varphi$ in the language either $T$ proves it or it proves its negation. From this you can prove that both the set of provable, and the set of non-provable are recursively enumerable, and therefore recursive.