It rarely makes sense to speak of formulas; the first quantifier of a formula is either existential or universal. So a set is not necessarily defined by a formula in the sense of a formula that is both and ; rather, there are both and formulas that define the set. For example, the set of odd natural numbers is definable by either or .

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the set of natural numbers. Instead of formulas with