following rules: if there is an i such that f(i) > 0, if there is no i such that f(i) > 0. Using a precise definition of computability relative to a type-2 functional, Kleene showed that a set of natural numbers is hyperarithmetical if and only if it is computable relative to . Example: the truth set of arithmetic Every arithmetical set is hyperarithmetical, but there are many other hyperarithmetical sets. One example of a hyperarithmetical, nonarithmetical set is the set T of Gödel numbers of formulas of Peano arithmetic that are true in the standard natural numbers . The set T is Turing equivalent to the set , and so is not high in the hyperarithmetical hierarchy, although it is not arithmetically definable by Tarski's indefinability theorem. Fundamental results The fundamental results of hyperarithmetic theory show that the three definitions above define the same collection of sets of natural numbers. These equivalences are due to Kleene. Completeness results are also fundamental to the theory. A set of natural numbers is complete if it is at level of the analytical hierarchy and every set of natural numbers is many-one reducible to it. The definition of a complete subset of Baire space () is similar. Several sets associated with hyperarithmetic theory are complete: Kleene's , the set of natural numbers that are notations for ordinal numbers The set of natural numbers e such that the computable function computes the characteristic function of a well ordering of the natural numbers. These are the indices of recursive ordinals. The set of elements of Baire space that are the characteristic functions of a well ordering of the natural numbers (using an effective isomorphism . Results known as bounding follow from these completeness results. For any set S of ordinal notations, there is an such that every element of S is a notation for an ordinal less than . For any subset T of Baire space consisting only of characteristic functions of well orderings, there is an such that each ordinal represented in T is less than . Relativized hyperarithmeticity and hyperdegrees The definition of can be relativized to a set X of natural numbers: in the definition of an ordinal notation, the clause for limit ordinals is changed so that the computable enumeration of a sequence of ordinal notations is allowed to use X as an oracle. The set of numbers that are ordinal