Define number quantifiers n,n, and n, such that for any language L and any L-structure A and any L-formula (x) : A(nx)A(x)A(nx)thereexistatmostnelementsaAsuchthatA[xa],thereexistatleastnelementsaAsuchthatA[xa],thereexistexactlynelementsaAsuchthatA[xa]. You can use the quantifier along with as a logical symbol, as well as ,,,,. Caution: The structure A need not contain any numbers, and possibly numbers cannot even be defined in the language L ! The cardinality n occurs metamathematically (such as in the indices of the variables - in fact, you will need at least n variables and accordant quantifiers)