analytic hierarchy
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:

$$\varphi (\overrightarrow{k})=\forall {X}_{1}\exists {X}_{2}\mathrm{\cdots}Q{X}_{n}\psi (\overrightarrow{k},{\overrightarrow{X}}_{n})$$ 


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