Editing Analytical hierarchy (section)

== The analytical hierarchy of sets of natural numbers ==
A set of natural numbers is assigned the classification <math>\Sigma^1_n</math> if it is definable by a <math>\Sigma^1_n</math> formula (with one free number variable and no free set variables). The set is assigned the classification <math>\Pi^1_n</math> if it is definable by a <math>\Pi^1_n</math> formula. If the set is both <math>\Sigma^1_n</math> and <math>\Pi^1_n</math> then it is given the additional classification <math>\Delta^1_n</math>.

The <math>\Delta^1_1</math> sets are called '''hyperarithmetical'''. An alternate classification of these sets by way of iterated computable functionals is provided by the [[hyperarithmetical theory]].