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
Famous quotes containing the words formal and/or definition:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)