Never mind, that’s trivial, you can win at that decision problem via the agent that does one thing if its proof search succeeds and another if its proof search fails.
To generalize, though, if there are N different actions, and for each action there’s a decision problem which rewards you iff PA+M proves you take that action, then I think that there exists a decision theory which wins at all of them iff M≥N−1. (I think you can inductively prove that the number of possible actions a decision theory can take if PA+m is inconsistent is ≤m+1.)
Never mind, that’s trivial, you can win at that decision problem via the agent that does one thing if its proof search succeeds and another if its proof search fails.
To generalize, though, if there are N different actions, and for each action there’s a decision problem which rewards you iff PA+M proves you take that action, then I think that there exists a decision theory which wins at all of them iff M≥N−1. (I think you can inductively prove that the number of possible actions a decision theory can take if PA+m is inconsistent is ≤m+1.)