In class theories, the axiom of limitation of size says that for any class C, C is a set (a class which can be an element of other classes) if and only if V (the class of all sets) cannot be mapped one-to-one into C.

This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, and axiom of global choice at one stroke. The axiom of limitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is an injection from the universe to the ordinals. Thus the universe of sets is well-ordered. John von Neumann in the 1940s. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... In class theories, the axiom of global choice is a stronger variant of the axiom of choice which applies to proper classes as well as sets. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...

See also

• Axiom of global choice
• Limitation of size
• Von Neumann–Bernays–Gödel set theory
• Morse–Kelley set theory