A theory is a set of first order sentences over some signature. A set of sentences are called axioms for the theory, if the deductive closure of the axioms equals the theory.
Now, if I have a recursively enumerable set of axioms, then the theory is recursively enumerable. For example we could enumerate up to the $i$-th axiom, compute all deductions of length $i$, then continue with $i+1$.
Do you know an example of a recursively enumerable theory, that does not has a decidable set of axioms?