Another opinion concerning Richard's paradox relates to mathematical predicativism. By this view, the real numbers are defined in stages, with each stage only making reference to previous stages and other things that have already been defined. From a predicative viewpoint it is not valid to quantify over "all" real numbers in the process of generating a new real number, because this is believed to result in a circularity problem in the definitions. Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.