Let $T$ be a complete extension of ZF. Is there an algorithm for extending $T$ to a complete extension of NBG? E.g., if $T$ is the complete theory of some $M \models ZF,$ is the complete theory of $\text{Def}(M)$ Turing reducible to $T?$ I think this is impossible by some sort of undefinability of truth argument, yet I can't come up with anything that can be done in $\text{Def}(M)$ which clearly cannot be simulated in $M.$

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    $\begingroup$ A trivial comment: it's crucial that we talk about extensions, since the Turing degrees of completions of ZF are exactly the Turing degrees of completions of NBG - that is, every completion $T_0$ of ZF computes a completion $T_1$ of NBG, and this is even uniformly true. Of course, this all turns into nonsense once we demand $T_0\subseteq T_1$ ... $\endgroup$ – Noah Schweber May 18 '18 at 4:18

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