In recursion 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 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 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 if it is both and . The hyperarithmetical sets are exactly the sets.
Hyperarithmetical sets and iterated Turing jumps: the hyperarithmetical hierarchy
The definition of hyperarithmetical sets as 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 . The number 0 is a notation for the ordinal 0. If n is a notation for an ordinal λ then is a notation for λ + 1; Suppose that δ is a limit ordinal. A notation for δ is a number of the form , where e is the index of a total computable function such that for each n, is a notation for an ordinal λn less than δ and δ is the sup of the set .
This may also be defined by taking effective joins at all levels instead of only notations for limit ordinals.
There are only countably many ordinal notations, since each notation is a natural number; thus there is a countable ordinal which is the supremum of all ordinals that have a notation. This ordinal is known as the Church-Kleene ordinal and is denoted . Note that this ordinal is still countable, the symbol being only an analogy with the first uncountable ordinal, . The set of all natural numbers that are ordinal notations is denoted and called Kleene's .
Ordinal notations are used to define iterated Turing jumps. The sets of natural numbers used to define the hierarchy are for each . is sometimes also denoted , or for a notation for . Suppose that δ has notation e. These sets were first defined by Davis (1950) and Mostowski (1951). The set is defined using e as follows. If δ = 0 then is the empty set. If δ = λ + 1 then is the Turing jump of . The sets and are and , respectively. If δ is a limit ordinal, let be the sequence of ordinals less than δ given by the notation e. The set is given by the rule . This is the effective join of the sets . Although the construction of depends on having a fixed notation for δ, and each infinite ordinal has many notations, a theorem of Spector shows that the Turing degree of depends only on δ, not on the particular notation used, and thus is well defined up to Turing degree.
The hyperarithmetical hierarchy is defined from these iterated Turing jumps. A set X of natural numbers is classified at level δ of the hyperarithmetical hierarchy, for , if X is Turing reducible to . There will always be a least such δ if there is any; it is this least δ that measures the level of uncomputability of X.