between it and V. The axiom of limitation of size implies von Neumann's 1923 axiom. Therefore, it also implies that all proper classes are equinumerous with V. Zermelo's models and the axiom of limitation of size 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. The model Vω To demonstrate that Theorems 1 and 2 hold for some Vκ, we first prove that if a set belongs to Vα then it belongs to all subsequent Vβ, or equivalently: Vα ⊆ Vβ for α ≤ β. This is