I'm relearning some computability theory, and have encountered the idea of relativization of results to arbitrary subsets of $\omega$ and the subtlety of figuring out what the correct relativized statement is. One way of figuring out the correct relativized statement is to go through the proof of the original statement and make sure that the same proof goes through, just relativized.
My question is: is there a way to simply look at a statement and know what the correct relativization is without knowing the proof? Also, is there a technical sense in which there a unique correct relativization?
