TITLE: Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?
QUESTION [57 upvotes]: To be more specific, does there exist a decision problem $P$ such that

given an oracle machine solving $P$, the Halting problem remains undecidable, and
given an oracle machine solving the Halting problem, $P$ remains undecidable?

REPLY [3 votes]: The Wikipedia article on Turing degree states that for every non-zero degree there is an incomparable degree. In your case, this means that there is some degree which is incomparable with 0', and any member of this degree answers your question.