Is it possible to demonstrate that a sentence must be formally independent based on the fact that it is non-relativizing? In other words, are there examples of sentences in computability/complexity theory where it can be demonstrated both a) that all proofs that resolve the question of whether or not two classes are equal must relativize, and b) that there are no relativizing proofs that can be used in such a resolution?
I think that results satisfying part b would be easier to come by. Another way to ask this question is: Has there ever been a sentence in computability or complexity theory where it can be demonstrated that the equality or inequality must be established through the use of (and only through the use of) relativizing techniques? An example of this would be interesting to me.
Thanks; an answer to either version of this question would be very interesting to me.