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Wikipedia / Hyperarithmetical theory

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 of this hierarchy if it is definable by a formula of second-order arithmetic sets.

Hyperarithmetical sets and iterated Turing jumps: the hyperarithmetical hierarchy

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 is a notation for + 1;

  • Suppose that is a limit ordinal. A notation for , where e such that for each n is a notation for an ordinal n less than and is the sup.

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 that is the supremum of all ordinals that have a notation. This ordinal is known as the Church–Kleene ordinal and called \mathcal{O}.

,. Suppose that has notation e. These sets were first defined by Davis (1950) and Mostowski (1951). is defined using e as follows.

  • If is the empty set.

  • If = , respectively.

  • If be the sequence of ordinals less than given by the notation e. This is the effective join.

depends on having a fixed notation for , and each infinite ordinal has many notations, a theorem of Clifford Spector shows that the Turing degree depends only on 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 , if X is Turing reducible. There will always be a least such if there is any; it is this least that measures the level of uncomputability of X.

Hyperarithmetical sets and constructibility

th level of the constructible hierarchy be the map from a member of Kleene's O is from the Lévy hierarchy of formulae.

.Theorem 3.7.1

Hyperarithmetical sets and recursion in higher types

A third characterization of the hyperarithmetical sets, due to Kleene, uses higher-type is defined by the following rules:

  1. if there is an i such that f(i) > 0,

  2. if there is no i such that f(i) > 0.

.

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. The set T is Turing equivalent, 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. Another characterization is given by the Suslin–Kleene theorem.

Completeness results are also fundamental to the theory. A set of natural numbers is complete of the analytical hierarchy set of natural numbers is many-one reducible complete:

  • \mathcal{O}, the set of natural numbers that are notations for ordinal numbers

  • The set of natural numbers e 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.

Results known as bounding set S such that every element of S subset T such that each ordinal represented in T.

Relativized hyperarithmeticity and hyperdegrees

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 notations relative to X.

is defined to be X is the Turing jump of X the hierarchy relative to X.

The relativized hyperarithmetical hierarchy is used to define hyperarithmetical reducibility. Given sets X and Y such that X 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 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.

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 .

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. , 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