Arithmetical set


In mathematical logic, an arithmetical 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.
The definition can be extended to an arbitrary countable set A by using Gödel numbers to represent elements of the set and declaring a subset of A to be arithmetical if the set of corresponding Gödel numbers is arithmetical.
A function is called arithmetically definable if the graph of is an arithmetical set.
A real number is called arithmetical if the set of all smaller rational numbers is arithmetical. A complex number is called arithmetical if its real and imaginary parts are both arithmetical.

Formal definition

A set X of natural numbers is arithmetical or arithmetically definable if there is a formula φ in the language of Peano arithmetic such that each number n is in X if and only if φ holds in the standard model of arithmetic. Similarly, a k-ary relation is arithmetical if there is a formula such that holds for all k-tuples 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

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 in the language of Peano arithmetic with no free number variables and a new set parameter Z and set membership relation such that Y is the unique set Z such that holds.
Every arithmetical set is implicitly arithmetical; if X is arithmetically defined by φ then it is implicitly defined by the formula
Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first-order arithmetic is implicitly arithmetical but not arithmetical.