  It might seem odd to refer to the above hierarchies as hierarchies of definability. However, as the notation indicates, the above hierarchies can be recast in terms of definability. For example, the projective sets are exactly the sets which are definable (with parameters) in second-order arithmetic. In the notation Σ̰1n, the superscript indicates that we are allowing quantification over objects of second-order (that is, sets of integers (or, equivalently, real numbers)), ‘Σ’ indicates that the starting quantifier is existential, the subscript indicates that there are n-alternations of quantifiers, and ‘∼’ indicates that we are allowing real parameters. In the case where real parameters are prohibited, the corresponding pointclasses are denoted Σ1n, Π1n, and Δ1n. The former hierarchy (where real parameters are allowed) is called a boldface hierarchy, while the latter (where real parameters are prohibited) is called a lightface hierarchy. In a similar manner one can define a lightface version of the Borel  hierarchy.[11]  Thus, we really have two hierarchies of definability: the lightface hierarchy and the boldface hierarchy. 