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. 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 (NBG) and Morse–Kelley set theory. Later expositions of class theories—such as those of Paul Bernays, Kurt Gödel, and John L. Kelley—use replacement, union, and a choice axiom equivalent to global choice rather than von Neumann's axiom. In 1930, Ernst Zermelo defined models of set theory satisfying the axiom of limitation of size. Abraham Fraenkel and Azriel Lévy have stated that the axiom of limitation of size does not capture all of the "limitation of size doctrine" because it does not imply the power set axiom. Michael Hallett has argued that the limitation of size doctrine does not justify the power set axiom and that "von Neumann's explicit assumption [of the smallness of power sets] seems preferable to Zermelo's, Fraenkel's, and Lévy's obscurely hidden implicit assumption of the smallness of power sets."

Von Neumann proved that the axiom of limitation of size implies the axiom of replacement, which can be expressed as: If F is a function and A is a set, then F(A) is a set. This is proved by contradiction. Let F be a function and A be a set. Assume that F(A) is a proper class. Then there is a function G that maps F(A) onto V. Since the composite function G ∘ F maps A onto V, the axiom of limitation of size implies that A is a proper class, which contradicts A being a set. Therefore, F(A) is a set. Since the axiom of replacement implies the axiom of separation, the axiom of limitation of size implies the axiom of separation. Von Neumann also proved that his axiom implies that V can be well ordered. The proof starts by proving by contradiction that Ord, the class of all ordinals, is a proper class. Assume that Ord is a set. Since it is a transitive set that is strictly well ordered by ∈, it is an ordinal. So Ord ∈ Ord, which contradicts Ord being strictly well ordered by ∈. Therefore, Ord is a proper class. So von Neumann's axiom implies that there is a function F that maps Ord onto V. To define a well ordering of V, let G be the subclass of F consisting of the ordered pairs (α, x) where α is the least β such that (β, x) ∈ F; that is, G = {(α, x) ∈ F: ∀β((β, x) ∈ F ⇒ α ≤ β)}. The function G is a one to one correspondence between a subset of Ord and V. Therefore, x < y if G−1(x) < G−1(y) defines a well ordering of V. This well ordering defines a global choice function: Let Inf(x) be the least element of a non empty set x. Since Inf(x) ∈ x, this function chooses an element of x for every non empty set x. Therefore, Inf(x) is a global choice function, so Von Neumann's axiom implies the axiom of global choice. In 1968, Azriel Lévy proved that von Neumann's axiom implies the axiom of union. First, he proved without using the axiom of union that every set of ordinals has an upper bound. Then he used a function that maps Ord onto V to prove that if A is a set, then ∪A is a set. The axioms of replacement, global choice, and union (with the other axioms of NBG) imply the axiom of limitation of size. Therefore, this axiom is equivalent to the combination of replacement, global choice, and union in NBG or Morse–Kelley set theory. These set theories only substituted the axiom of replacement and a form of the axiom of choice for the axiom of limitation of size because von Neumann's axiom system contains the axiom of union. Lévy's proof that this axiom is redundant came many years later. The axioms of NBG with the axiom of global choice replaced by the usual axiom of choice do not imply the axiom of limitation of size. In 1964, William B. Easton used forcing to build a model of NBG with global choice replaced by the axiom of choice. In Easton's model, V cannot be linearly ordered, so it cannot be well ordered. Therefore, the axiom of limitation of size fails in this model. Ord is an example of a proper class that cannot be mapped onto V because (as proved above) if there is a function mapping Ord onto V, then V can be well ordered. The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size. Define as the th infinite initial ordinal, which is also the cardinal ; numbering starts at , so In 1939, Gödel pointed out that Lωω, a subset of the constructible universe, is a model of ZFC with replacement replaced by separation. To expand it into a model of NBG with replacement replaced by separation, let its classes be the sets of Lωω+1, which are the constructible subsets of Lωω. This model satisfies NBG's class existence axioms because restricting the set variables of these axioms to Lωω produces instances of the axiom of separation, which holds in L. It satisfies the axiom of global choice because there is a function belonging to Lωω+1 that maps ωω onto Lωω, which implies that Lωω is well ordered. The axiom of limitation of size fails because the proper class {ωn : n ∈ ω} has cardinality , so it cannot be mapped onto Lωω, which has cardinality
In a 1923 letter to Zermelo, von Neumann stated the first version of his axiom: A class is a proper class if and only if there is a one to one correspondence between it and V.
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.

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 proved by transfinite induction on β:
β = 0: V0 ⊆ V0. For β+1: By inductive hypothesis, Vα ⊆ Vβ. Hence, Vα ⊆ Vβ ⊆ Vβ ∪ P(Vβ) = Vβ+1. For limit β: If α < β, then Vα ⊆ ∪ξ < β Vξ = Vβ. If α = β, then Vα ⊆ Vβ. Sets enter the cumulative hierarchy through the power set P(Vβ) at step β+1. The following definitions will be needed:
If x is a set, rank(x) is the least ordinal β such that x ∈ Vβ+1. The supremum of a set of ordinals A, denoted by sup A, is the least ordinal β such that α ≤ β for all α ∈ A.

Zermelo's smallest model is Vω. Mathematical induction proves that Vn is finite for all n < ω:
|V0| = 0. |Vn+1| = |Vn ∪ P(Vn)| ≤ |Vn| + 2 |Vn|, which is finite since Vn is finite by inductive hypothesis. Proof of Theorem 1: A set X enters Vω through P(Vn) for some n < ω, so X ⊆ Vn. Since Vn is finite, X is finite. Conversely: If a class X is finite, let N = sup {rank(x): x ∈ X}. Since rank(x) ≤ N for all x ∈ X, we have X ⊆ VN+1, so X ∈ VN+2 ⊆ Vω. Therefore, X ∈ Vω. Proof of Theorem 2: Vω is the union of countably infinitely many finite sets of increasing size. Hence, it has cardinality , which equals ω by von Neumann cardinal assignment. The sets and classes of Vω satisfy all the axioms of NBG except the axiom of infinity.