. The converse of this implication is not true. (Shore and Slaman, 1999) The function mapping to is definable in the partial order of the Turing degrees. Many properties of the Turing jump operator are discussed in the article on Turing degrees. References Ambos-Spies, K. and Fejer, P. Degrees of Unsolvability. Unpublished. http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf Year of introduction missing Computability theory Alan Turing