analytic hierarchy

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 ${\mathrm{\Delta }}_0^1$, ${\mathrm{\Delta }}_1^1$, ${\mathrm{\Sigma }}_0^1$, or ${\mathrm{\Pi }}_0^1$, and consists of the arithmetical formulas or relations.

A formula $\varphi$ is ${\mathrm{\Sigma }}_n^1$ if there is some arithmetical formula $\psi$ such that:

$$\varphi (\overrightarrow{k})=\exists X_1\forall X_2\mathrm{\cdots }Q{X}_n\psi (\overrightarrow{k},{\overrightarrow{X}}_n)$$

$$\text{ where }Q\text{ is either }\forall \text{ or }\exists \text{, whichever maintains the pattern of alternating quantifiers, and each }{X}_i\text{ is a set variable (that is, second order)}$$

Similarly, a formula $\varphi$ is ${\mathrm{\Pi }}_n^1$ if there is some arithmetical formula $\psi$ such that:

$$\varphi (\overrightarrow{k})=\forall X_1\exists X_2\mathrm{\cdots }Q{X}_n\psi (\overrightarrow{k},{\overrightarrow{X}}_n)$$

$$\text{ where }Q\text{ is either }\forall \text{ or }\exists \text{, whichever maintains the pattern of alternating quantifiers, and each }{X}_i\text{ is a set variable (that is, second order)}$$