Arithmetical set

In mathematical logic, an arithmetical set (or arithmetic set) is a set of natural numbers that can be defined by a formula of first-order Peano arithmetic. The arithmetical sets are classified by the arithmetical hierarchy.

A function $f:subseteq\; mathbb\{N\}^k\; o\; mathbb\{N\}$ is called arithmetically definable if the graph of $f$ is an arithmetical set.

Formal definition

A set "X" of natural numbers is arithmetical or arithmetically definable if there is a formula φ("n") in the language of Peano arithmetic such that each number "n" is in "X" if and only if φ("n") holds in the standard model of arithmetic. Similarly, a "k"-ary relation$R(n\_1,ldots,n\_k)$ is arithmetical if there is a formula $psi(n\_1,ldots,n\_k)$ such that $R(n\_1,ldots,n\_k)\; Leftrightarrow\; psi(n\_1,ldots,n\_k)$ holds for all "k"-tuples $(n\_1,ldots,n\_k)$ of natural numbers.

A finitary function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation.

A set "A" is said to be arithmetical in a set "B" if "A" is definable by an arithmetical formula which has "B" as a set parameter.

Examples

* Every computable function is arithmetically definable.
* The set of all prime numbers is arithmetical.
* Every recursively enumerable set is arithmetical.
* The set encoding the Halting problem is arithmetical.
* Chaitin's constant Ω is encoded by an arithmetical set.
* Tarski's indefinability theorem shows that the set of Gödel numbers of true formulas of first order arithmetic is not arithmetically definable.

Properties

* The complement of an arithmetical set is an arithmetical set.
* The Turing jump of an arithmetical set is an arithmetical set.
* The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumerates all arithmetical sets.

Implicitly arithmetical sets

Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property.

A set "Y" of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable with an arithmetical formula that is able to use "Y" as a parameter. That is, if there is a formula $heta(Z)$ in the language of Peano arithmetic with no free number variables and a new set parameter "Z" and set membership relation $in$ such that "Y" is the unique set such that $heta(Y)$ holds.

Every arithmetical set is implicitly arithmetical; if "X" is arithmetically defined by φ("n") then it is implicitly defined by the formula:$forall\; n\; [n\; in\; Z\; Leftrightarrow\; phi(n)]$.Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.

See also

* Arithmetical hierarchy
* Computable set
* Computable number

References

Rogers, H. (1967). "Theory of recursive functions and effective computability." McGraw-Hill.