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.
The definition can be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language, etc.)
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.
is called arithmetically definable if the graph of
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.
A set X of natural numbers is arithmetical or arithmetically definable if there is a first-order 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
holds for all k-tuples
is called arithmetical if its graph is an arithmetical (k+1)-ary relation.
A set A is said to be arithmetical in a set B if A is definable by an arithmetical formula that has B as a set parameter.
Each arithmetical set has an arithmetical formula that says whether particular numbers are in the set.
An alternative notion of definability allows for a formula that does not say whether particular numbers are in the set but says 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.
in the language of Peano arithmetic with no free number variables and a new set parameter Z and set membership relation
Every arithmetical set is implicitly arithmetical; if X is arithmetically defined by φ(n) then it is implicitly defined by the formula Not every implicitly arithmetical set is arithmetical, however.