Consider the first-order language $\mathcal{L}_{\text{OA}}:=(+,\cdot,0,1)$; in this language, we can formulate statements of ordinal arithmetic. Clearly, the theory $T_{\text{OA}}$ of $(\text{On},+,\cdot,0,1)$ is not recursive, as $\omega$ is definable in $\mathcal{L}_{\text{OA}}$ (as being the only ordinal $\gamma$ different from $0$, not having a predecessor and having no $\alpha$ and $\beta$ different from $0$ such that $\alpha+\beta=\gamma$ with $\alpha$ not having a predecessor) and hence the theory TA (true arithmetic) of $\mathbb{N}$ is computable from $T_{\text{OA}}$. Note that $T_{\text{OA}}$ is absolute between transitive class models of ZFC.

My question is whether the reverse reduction holds: Can we decide $T_{\text{OA}}$, given TA? More precisely, what is the Turing degree of $T_{\text{OA}}$?

Linear Orderingscontains a proof that this is just $T_{OA}(\alpha)$ for some specific countable indecomposable $\alpha$. On the other hand, with a lot of extra functions, you get Silver machines that encode $L$ as @JoelDavidHamkins suggests. $\endgroup$ – François G. Dorais♦ Mar 7 '16 at 21:46