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."

Formal statement

The usual version of the axiom of limitation of size—a class is a proper class if and only if there is a function that maps it onto V—is expressed in the formal language of set theory as:

Gödel introduced the convention that uppercase variables range over all the classes, while lowercase variables range over all the sets. This convention allows us to write:

• instead of
• instead of

With Gödel's convention, the axiom of limitation of size can be written:

Implications of the axiom

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&nbsp;∈&nbsp;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,&nbsp;x) where α is the least β such that (β,&nbsp;x)&nbsp;∈&nbsp;F; that is, G&nbsp;=&nbsp;{(α,&nbsp;x)&nbsp;∈&nbsp;F:&nbsp;∀β((β,&nbsp;x)&nbsp;∈&nbsp;F&nbsp;⇒&nbsp;α&nbsp;≤&nbsp;β)}. The function G is a one-to-one correspondence between a subclass of Ord and V. Therefore, x&nbsp;<&nbsp;y if G^{−1}(x)&nbsp;<&nbsp;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)&nbsp;∈&nbsp;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 L_{ω_ω}, 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 L_{ω_ω+1}, which are the constructible subsets of L_{ω_ω}. This model satisfies NBG's class existence axioms because restricting the set variables of these axioms to L_{ω_ω} 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 L_{ω_ω+1} that maps ω_ω onto L_{ω_ω}, which implies that L_{ω_ω} is well-ordered. The axiom of limitation of size fails because the proper class {ω_n&nbsp;:&nbsp;n&nbsp;∈&nbsp;ω} has cardinality , so it cannot be mapped onto L_{ω_ω}, 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. 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 V_α, which is defined by transfinite recursion:

1. V₀&nbsp;=&nbsp;âˆÂ.
2. V_α+1&nbsp;=&nbsp;V_α&nbsp;∪&nbsp;P(V_α). That is, the union of V_α and its power set.
3. For limit β: V_β&nbsp;=&nbsp;∪_{α&nbsp;<&nbsp;β}&nbsp;V_α. That is, V_β is the union of the preceding V_α.

Zermelo worked with models of the form V_κ where κ is a cardinal. The classes of the model are the subsets of V_κ, and the model's ∈-relation is the standard ∈-relation. The sets of the model are the classes X such that X ∈ V_κ. Zermelo identified cardinals κ such that V_κ satisfies:

Theorem 1. A class X is a set if and only if |X|&nbsp;<&nbsp;κ.

Theorem 2. |V_κ|&nbsp;=&nbsp;κ.

Since every class is a subset of V_κ, Theorem 2 implies that every class X has cardinality&nbsp;≤&nbsp;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 V_κ. This correspondence is a subset of V_κ, so it is a class of the model. Therefore, the axiom of limitation of size holds for the model V_κ.

The theorem stating that V_κ has a well-ordering can be proved directly. Since κ is an ordinal of cardinality κ and |V_κ|&nbsp;=&nbsp;κ, there is a one-to-one correspondence between κ and V_κ. This correspondence produces a well-ordering of V_κ. 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 V_κ.

The model V_ω

To demonstrate that Theorems 1 and 2 hold for some V_κ, we first prove that if a set belongs to V_α then it belongs to all subsequent V_β, or equivalently: V_α&nbsp;⊆&nbsp;V_β for α&nbsp;≤&nbsp;β. This is proved by transfinite induction on β:

1. β&nbsp;=&nbsp;0: V₀&nbsp;⊆&nbsp;V₀.
2. For β+1: By inductive hypothesis, V_α&nbsp;⊆&nbsp;V_β. Hence, V_α&nbsp;⊆&nbsp;V_β&nbsp;⊆&nbsp;V_β&nbsp;∪&nbsp;P(V_β)&nbsp;=&nbsp;V_β+1.
3. For limit β: If α&nbsp;<&nbsp;β, then V_α&nbsp;⊆&nbsp;∪_{ξ&nbsp;<&nbsp;β}&nbsp;V_ξ&nbsp;=&nbsp;V_β. If α&nbsp;=&nbsp;β, then V_α&nbsp;⊆&nbsp;V_β.

Sets enter the cumulative hierarchy through the power set P(V_β) 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&nbsp;∈&nbsp;V_β+1.

The supremum of a set of ordinals A, denoted by sup&nbsp;A, is the least ordinal β such that α&nbsp;≤&nbsp;β for all α&nbsp;∈&nbsp;A.

Zermelo's smallest model is V_ω. Mathematical induction proves that V_n is finite for all n&nbsp;<&nbsp;ω:

1. |V₀|&nbsp;=&nbsp;0.
2. |V_n+1|&nbsp;=&nbsp;|V_n&nbsp;∪&nbsp;P(V_n)|&nbsp;≤&nbsp;|V_n|&nbsp;+&nbsp;2&nbsp;^|V_n|, which is finite since V_n is finite by inductive hypothesis.

Proof of Theorem 1: A set X enters V_ω through P(V_n) for some n&nbsp;<&nbsp;ω, so X&nbsp;⊆&nbsp;V_n. Since V_n is finite, X is finite. Conversely: If a class X is finite, let N&nbsp;=&nbsp;sup&nbsp;{rank(x):&nbsp;x&nbsp;∈&nbsp;X}. Since rank(x)&nbsp;≤&nbsp;N for all x&nbsp;∈&nbsp;X, we have X&nbsp;⊆&nbsp;V_N+1, so X&nbsp;∈&nbsp;V_N+2&nbsp;⊆&nbsp;V_ω. Therefore, X&nbsp;∈&nbsp;V_ω.

Proof of Theorem 2: V_ω 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 V_ω satisfy all the axioms of NBG except the axiom of infinity.

The models V_κ where κ is a strongly inaccessible cardinal

Two properties of finiteness were used to prove Theorems 1 and 2 for V_ω:

1. If λ is a finite cardinal, then 2^λ is finite.
2. If A is a set of ordinals such that |A| is finite, and α is finite for all α&nbsp;∈&nbsp;A, then sup&nbsp;A is finite.

To find models satisfying the axiom of infinity, replace "finite" by "<&nbsp;κ" to produce the properties that define strongly inaccessible cardinals. A cardinal κ is strongly inaccessible if κ&nbsp;>&nbsp;ω and:

1. If λ is a cardinal such that λ&nbsp;<&nbsp;κ, then 2^λ&nbsp;<&nbsp;κ.
2. If A is a set of ordinals such that |A|&nbsp;<&nbsp;κ, and α&nbsp;<&nbsp;κ for all α&nbsp;∈&nbsp;A, then sup&nbsp;A&nbsp;<&nbsp;κ.

These properties assert that κ cannot be reached from below. The first property says κ cannot be reached by power sets; the second says κ cannot be reached by the axiom of replacement. Just as the axiom of infinity is required to obtain ω, an axiom is needed to obtain strongly inaccessible cardinals. Zermelo postulated the existence of an unbounded sequence of strongly inaccessible cardinals.

If κ is a strongly inaccessible cardinal, then transfinite induction proves |V_α|&nbsp;<&nbsp;κ for all α&nbsp;<&nbsp;κ:

1. α&nbsp;=&nbsp;0: |V₀|&nbsp;=&nbsp;0.
2. For α+1: |V_α+1|&nbsp;=&nbsp;|V_α&nbsp;∪&nbsp;P(V_α)|&nbsp;≤ |V_α|&nbsp;+&nbsp;2&nbsp;^|V_α|&nbsp;=&nbsp;2&nbsp;^|V_α|&nbsp;<&nbsp;κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.
3. For limit α: |V_α|&nbsp;=&nbsp;|∪_{ξ&nbsp;<&nbsp;α}&nbsp;V_ξ|&nbsp;≤ sup&nbsp;