 The paradoxes of naive set theory can be explained in terms of the inconsistent tacit assumption that "all classes are sets".With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets).For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.