of a set is a form of the halting problem relative to . Given a set , the Turing jump is the set of indices of oracle Turing machines that halt on input when run with oracle . It is known that every set is Turing reducible to its Turing jump, but the Turing jump of a set is never Turing reducible to the original set.

Post's theorem uses finitely iterated Turing jumps. For any set of natural numbers, the notation indicates the -fold iterated