 A class function is not a function in the usual sense, since it is not a set; it is rather a formula formula_12 with the property that for any set formula_13 there is no more than one set formula_14 such that the pair formula_15 satisfies formula_16 For example, the class function mapping each set to its successor may be expressed as the formula formula_17 The fact that the ordered pair formula_15 satisfies formula_19 may be expressed with the shorthand notation formula_20

Another approach is taken by the von Neumann–Bernays–Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class.However, the class existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes.This causes NBG to be a conservative extension of ZF.