Axiom of limitation of size

In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. This axiom formalizes the limitation of size principle, which avoids the paradoxes by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is the member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V — that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.

Von Neumann's axiom implies the axioms of replacement, separation, global choice, and union. It is equivalent to the combination of replacement, global choice, and union in Von Neumann–Bernays–Gödel set theory (NBG) or Morse–Kelley set theory. Later expositions of class theories—such as those of Paul Bernays, Kurt Gödel, and John L. Kelley—use replacement and a form of the axiom of choice rather than von Neumann's axiom. In 1930, Ernst Zermelo defined models of set theory satisfying the axiom of limitation of size.

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