notations relative to X is denoted . The supremum of ordinals represented in is denoted ; this is a countable ordinal no smaller than . The definition of can also be relativized to an arbitrary set of natural numbers. The only change in the definition is that is defined to be X rather than the empty set, so that is the Turing jump of X, and so on. Rather than terminating at the hierarchy relative to X runs through all ordinals less than . The relativized hyperarithmetical hierarchy is used to define hyperarithmetical reducibility. Given sets X and Y, we say if and only if there is a such that X is Turing reducible to . If and then the notation is used to indicate X and Y are hyperarithmetically equivalent. This is a coarser equivalence relation than Turing equivalence; for example, every set of natural numbers is hyperarithmetically equivalent to its Turing jump but not Turing equivalent to its Turing jump. The equivalence classes of hyperarithmetical equivalence are known as hyperdegrees. The function that takes a set X to is known as the hyperjump by analogy with the Turing jump. Many properties of the hyperjump and hyperdegrees have been established. In particular, it is known that Post's problem for hyperdegrees has a positive answer: for every set X of natural numbers there is a set Y of natural numbers such that . Generalizations Hyperarithmetical theory is generalized by α-recursion theory, which is the study of definable subsets of admissible ordinals. Hyperarithmetical theory is the special case in which α is . Relation to other hierarchies References H. Rogers, Jr., 1967. The Theory of Recursive Functions and Effective Computability, second edition 1987, MIT Press. (paperback), G. Sacks, 1990. Higher Recursion Theory, Springer-Verlag. S. Simpson, 1999. Subsystems of Second Order Arithmetic, Springer-Verlag. C. J. Ash, J. F. Knight, 2000. Computable Structures and the Hyperarithmetical Hierarchy, Elsevier. Citations External links Descriptive set theory. Notes by David Marker, University of Illinois at Chicago. 2002. Mathematical Logic II. Notes by Dag Normann, The University of Oslo. 2005. Antonio Montalbán: University of California, Berkeley and YouTube content creator Computability theory Hierarchy