Let's assume we are working in $(\mathbb{N}, +, \dot\ , 0,1)$. Let $T$ be a set of formulae that is closed under $\neg$ and such that the set of Godel numbers of formulae in $T$ is recursive.
Moreover, let $\mathcal{A}$ be an axiomatic system that is valid (it only proves sentences $\varphi$ such that $\mathbb{N} \models \varphi$), and effective, and assume $\mathcal{A}$ is such that any true sentence in $T$ is also a theorem of $\mathcal{A}$.
I need to show that the set of true sentences of $T$ (their Godel numbers, technically speaking) is a recursive set.
Here's what I'm thinking, but I'm not sure about the details:
Since $\mathcal{A}$ is efective, any of its theorems can be obtained recursively, so, moreover, the set of true sentences of $T$ is then a recursively enumerable, which is half the battle towards showing it's recursive. Now, I'm not sure my reasoning is valid there, but even if it is, I still have to show that its complement is also recursively enumerable. I'm assuming somehow the fact that $T$ is closed under $\neg$ helps.
How do I proceed from here?