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 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.
Read more about this topic: Arithmetical Set
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