page_title: Axiom of limitation of size
section_title: Zermelo's models and the axiom of limitation of size
hierarchical_section_title: Axiom of limitation of size / Zermelo's models and the axiom of limitation of size
caption_attribution_description: Ernst Zermelo
context_page_description: In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory 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 a 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, union, and global choice. It is equivalent to the combination of replacement, union, and global choice in Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory. Later expositions of class theories—such as those of Paul Bernays, Kurt Gödel, and John L.
context_section_description: In 1930, Zermelo published an article on models of set theory, in which he proved that some of his models satisfy the axiom of limitation of size. These models are built in ZFC by using the cumulative hierarchy Vα, which is defined by transfinite recursion:
  V₀ = ∅.
  Vα+1 = Vα ∪ P(Vα). That is, the union of Vα and its power set.
  For limit β: Vβ = ∪α < β Vα. That is, Vβ is the union of the preceding Vα.
  Zermelo worked with models of the form Vκ where κ is a cardinal. The classes of the model are the subsets of Vκ, and the model's ∈-relation is the standard ∈-relation. The sets of the model are the classes X such that X ∈ Vκ. Zermelo identified cardinals κ such that Vκ satisfies:
  Theorem 1. A class X is a set if and only if |X| < κ.
  Theorem 2. |Vκ| = κ.
  Since every class is a subset of Vκ, Theorem 2 implies that every class X has cardinality ≤ κ. Combining this with Theorem 1 proves: every proper class has cardinality κ. Hence, every proper class can be put into one-to-one correspondence with Vκ. This correspondence is a subset of Vκ, so it is a class of the model. Therefore, the axiom of limitation of size holds for the model Vκ.
  The theorem stating that Vκ has a well-ordering can be proved directly. Since κ is an ordinal of cardinality κ and |Vκ| = κ, there is a one-to-one correspondence between κ and Vκ. This correspondence produces a well-ordering of Vκ. Von Neumann's proof is indirect. It uses the Burali-Forti paradox to prove by contradiction that the class of all ordinals is a proper class. Hence, the axiom of limitation of size implies that there is a function that maps the class of all ordinals onto the class of all sets. This function produces a well-ordering of Vκ.
caption: Ernst Zermelo in the 1900s
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