In computability theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory. The central focus of hyperarithmetic theory is the sets of natural numbers known as hyperarithmetic sets. There are three equivalent ways of defining this class of sets; the study of the relationships between these different definitions is one motivation for the study of hyperarithmetical theory.
Hyperarithmetical sets and definability The first definition of the hyperarithmetic sets uses the analytical hierarchy. A set of natural numbers is classified at level Σ 1 1 {\displaystyle \Sigma _{1}^{1}} of this hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified at level Π 1 1 {\displaystyle \Pi _{1}^{1}} of the analytical hierarchy if it is definable by a formula of second-order arithmetic with only universal set quantifiers and no other set quantifiers. A set is Δ 1 1 {\displaystyle \Delta _{1}^{1}} if it is both Σ 1 1 {\displaystyle \Sigma _{1}^{1}} and Π 1 1 {\displaystyle \Pi _{1}^{1}} . The hyperarithmetical sets are exactly the Δ 1 1 {\displaystyle \Delta _{1}^{1}} sets.
Hyperarithmetical sets and iterated Turing jumps: the hyperarithmetical hierarchy The definition of hyperarithmetical sets as Δ 1 1 {\displaystyle \Delta _{1}^{1}} does not directly depend on computability results. A second, equivalent, definition shows that the hyperarithmetical sets can be defined using infinitely iterated Turing jumps. This second definition also shows that the hyperarithmetical sets can be classified into a hierarchy extending the arithmetical hierarchy; the hyperarithmetical sets are exactly the sets that are assigned a rank in this hierarchy. Each level of the hyperarithmetical hierarchy is indexed by a countable ordinal number (ordinal), but not all countable ordinals correspond to a level of the hierarchy. The ordinals used by the hierarchy are those with an ordinal notation, which is a concrete, effective description of the ordinal. An ordinal notation is an effective description of a countable ordinal by a natural number. A system of ordinal notations is required in order to define the hyperarithmetic hierarchy. The fundamental property an ordinal notation must have is that it describes the ordinal in terms of smaller ordinals in an effective way. The following inductive definition is typical; it uses a pairing function ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } .
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