The analytic hierarchy is a hierarchy of either (depending on context) formulas or relations similar to the arithmetical hierarchy. It is essentially the second order equivalent. Like the arithmetical hierarchy, the relations in each level are exactly the relations defined by the formulas of that level.

The first level can be called $\Delta^{1}_{0}$, $\Delta^{1}_{1}$, $\Sigma^{1}_{0}$, or $\Pi^{1}_{0}$, and consists of the arithmetical formulas or relations.

A formula $\phi$ is $\Sigma^{1}_{n}$ if there is some arithmetical formula $\psi$ such that:

Similarly, a formula $\phi$ is $\Pi^{1}_{n}$ if there is some arithmetical formula $\psi$ such that: