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.
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.
A function formula_1 is called arithmetically definable if the graph of formula_2 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("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 formula_3 is arithmetical if there is a formula formula_4 such that formula_5 holds for all "k"-tuples formula_6 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.
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 formula_7 in the language of Peano arithmetic with no free number variables and a new set parameter "Z" and set membership relation formula_8 such that "Y" is the unique set "Z" such that formula_7 holds.
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. In particular, the truth set of first-order arithmetic is implicitly arithmetical but not arithmetical.