Analytical hierarchy

In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy.

The notation $\Sigma _{0}^{1}=\Pi _{0}^{1}=\Delta _{0}^{1}$ indicates on the one hand the class of formulas that can be expressed as formulas of arbitrary [[finite\\ length of alternating universal and existential quantifiers for individuals over predicates linked by sentential connectives, and on the other the class of Borel sets.

A $\Sigma _{1}^{1}$ formula is a formula of the form $\exists X\phi$ where X is now a predicate and $\phi \in \Sigma _{1}^{1}$, while a $\Sigma _{1}^{1}$ set is a set of the form $\{x:(\exists y\in S)\ R(x,y)\}$, where S is Borel and R is a relation. A $\Sigma _{1}^{1}$ set can thus be seen as a projection of a Borel set.

A $\Sigma _{n+1}^{1}$ formula is a formula of the form $\exists X\phi$ where X is a predicate and $\phi \in \Sigma _{n}^{1}$; analogously, a $\Sigma _{n+1}^{1}$ set is a set of the form $\{x:(\exists y\in S)\ R(x,y)\}$, where S is $\Sigma _{n}^{1}$.

A $\Pi _{n}^{1}$ formula is the negation of a $\Sigma _{n}^{1}$ formula, and a $\Pi _{n}^{1}$ set is the complement of a $\Sigma _{n}^{1}$ set.

A formula or set is called $\Delta _{n}^{1}$ if it's both $\Sigma _{n}^{1}$ and $\Pi _{n}^{1}$.