Analytical hierarchy

In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, ${\displaystyle \mathbb {N} }$, and over functions from ${\displaystyle \mathbb {N} }$ to ${\displaystyle \mathbb {N} }$. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy.