The study of arbitrary (not necessarily recursively enumerable) Turing degrees involves the study of the Turing jump. Given a set "A", the "Turing jump" of "A" is a set of natural numbers encoding a solution to the halting problem for oracle Turing machines running with oracle "A". The Turing jump of any set is always of higher Turing degree than the original set, and a theorem of Friedburg shows that any set that computes the Halting problem can be obtained as the Turing jump of another set. Post's theorem establishes a close relationship between the Turing jump operation and the arithmetical hierarchy, which is a classification of certain subsets of the natural numbers based on their definability in arithmetic.