User:RJGray/Sandboxc

In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class which is not a set (an element of other classes), if and only if it can be mapped onto the class V of all sets.


 * $$\forall C \biggl( \lnot \exist W \left( C \in W \right) \iff \exist F \Bigl( \forall x \bigl( \exist W \left( x \in W \right) \Rightarrow \exist s \left( s \in C \land \langle s, x \rangle \in F \right) \bigr) \land \forall x \forall y \forall s \bigl( \left( \langle s, x \rangle \in F \land \langle s, y \rangle \in F \right) \Rightarrow x = y \bigr) \Bigr) \biggr).$$

This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union 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 a surjection from the ordinals to the universe, thus an injection from the universe to the ordinals, that is, the universe of sets is well-ordered.

Together the axiom of replacement and the axiom of global choice (with the other axioms of von Neumann–Bernays–Gödel set theory) imply this axiom. This axiom is thus equivalent to the combination of replacement, global choice, specification and union in von Neumann–Bernays–Gödel or Morse–Kelley set theory.

However, the axiom of replacement and the usual axiom of choice (with the other axioms of von Neumann–Bernays–Gödel set theory) do not imply von Neumann's axiom. In 1964, Easton used forcing to build a model that satisfies the axioms of von Neumann–Bernays–Gödel set theory with one exception: the axiom of global choice is replaced by the axiom of choice. In Easton's model, the axiom of limitation of size fails dramatically: the universe of sets cannot even be linearly ordered.

From this axiom, it can be shown that a class is a proper class if and only if it is equinumerous to V, but von Neumann's axiom does not capture all of the "limitation of size doctrine", because the axiom of power set is not a consequence of it. Later expositions of class theories (Bernays, Gödel, Kelley, ...) generally use replacement and a form of the axiom of choice rather than the axiom of limitation of size.

History
Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z [ZFC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments."

The historical development of the ZFC axioms began in 1908 when Zermelo chose axioms to support his proof of the well-ordering theorem and to avoid contradictory sets. In 1922, Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, … } where Z0 is the set of natural numbers, and Zn+1 is the power set of Zn. They also introduced the axiom of replacement, which guarantees the existence of this set. However, adding axioms as they are needed neither guarantees the existence of all reasonable sets nor clarifies the difference between sets that are safe to use and collections that lead to contradictions.

In a 1923 letter to Zermelo, von Neumann outlined an approach to set theory that identifies the sets that are "too big" (now called proper classes) and that can lead to contradictions. Von Neumann identified these sets using the criterion: "A set is 'too big' if and only if it is equivalent to the set of all things." He then restricted how these sets may be used: "… in order to avoid the paradoxes those [sets] which are 'too big' are declared to be impermissible as elements." By combining this restriction with his criterion, von Neumann obtained the axiom of limitation of size (which in the language of classes states): A class X is not an element of any class if and only if X is equivalent to the class of all sets. So von Neumann identified sets as classes that are not equivalent to the class of all sets. Von Neumann realized that, even with his new axiom, his set theory does not fully characterize sets.

Gödel found von Neumann's axiom to be "of great interest":


 * "In particular I believe that his [von Neumann's] necessary and sufficient condition which a property must satisfy, in order to define a set, is of great interest, because it clarifies the relationship of axiomatic set theory to the paradoxes. That this condition really gets at the essence of things is seen from the fact that it implies the axiom of choice, which formerly stood quite apart from other existential principles.  The inferences, bordering on the paradoxes, which are made possible by this way of looking at things, seem to me, not only very elegant, but also very interesting from the logical point of view. Moreover I believe that only by going farther in this direction, i.e., in the direction opposite to constructivism, will the basic problems of abstract set theory be solved."

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:
 * 1) V0 = &empty;.
 * 2) Vα+1 = Vα ∪ P(Vα). That is, the union of Vα and its power set.
 * 3) 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 Vκ 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 |&thinsp;X&thinsp;| < κ.
 * Theorem 2. |&thinsp;Vκ&thinsp;| = 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 proof of the axiom of global choice in Vκ is more direct than von Neumann's proof. First note that κ (being a von Neumann cardinal) is a well-ordered class of cardinality κ. Since Theorem 2 states that Vκ has cardinality κ, there is a one-to-one correspondence between κ and Vκ. This correspondence produces a well-ordering of Vκ, which implies the axiom of global choice. Von Neumann uses the Burali-Forti paradox to prove by contradiction that the class of all ordinals is a proper class, and then he applies the axiom of limitation of size to well-order the universal class.

The model Vω
To demonstrate that Theorems 1 and 2 hold for some Vκ, we need to prove that if a set belongs to Vα then it belongs to all subsequent Vβ, or equivalently: Vα &sube; Vβ for α ≤ β. This is proved by transfinite induction on β: Note that sets enter the hierarchy only through the power set P(Vβ) at step β+1. We will need the following definitions:
 * 1) β = 0: V0 &sube; V0.
 * 2) For β+1: By inductive hypothesis, Vα &sube; Vβ. Hence, Vα &sube; Vβ &sube; Vβ ∪ P(Vβ) = Vβ+1.
 * 3) For limit β: If α < β, then Vα &sube; ∪ξ < β Vξ = Vβ. If α = β, then Vα &sube; Vβ.
 * 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. Induction proves that Vn is finite for all n < ω:
 * |&thinsp;V0&thinsp;| = 0.
 * |&thinsp;Vn+1&thinsp;| = |&thinsp;Vn ∪ P(Vn)&thinsp;| ≤ |&thinsp;Vn&thinsp;| + 2 undefined, which is finite since Vn is finite by inductive hypothesis.

To prove Theorem 1: since a set X enters Vω only through P(Vn) for some n < ω, we have X &sube; 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 &sube; VN+1, so X ∈ VN+2 &sube; Vω. Therefore, X ∈ Vω.

To prove Theorem 2, note that Vω is the union of countably many finite sets. Hence, Vω is countably infinite and has cardinality $$\aleph_0$$ (which equals ω by von Neumann cardinal assignment).

It can be shown that the sets and classes of Vω satisfy all the axioms of NBG (von Neumann–Bernays–Gödel set theory) except the axiom of infinity.

The models Vκ where κ is a strongly inaccessible cardinal
To find models satisfying the axiom of infinity, observe that two properties of finiteness were used to prove Theorems 1 and 2 for Vω: Replacing "finite" by "< κ" produces the properties that define strongly inaccessible cardinals. A cardinal κ is strongly inaccessible if κ > ω and: 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.
 * 1) If λ is a finite cardinal, then 2λ is finite.
 * 2) If A is a set of ordinals such that |&thinsp;A&thinsp;| is finite, and α is finite for all α ∈ A, then sup A is finite.
 * 1) If λ is a cardinal such that λ < κ, then 2λ < κ.
 * 2) If A is a set of ordinals such that |&thinsp;A&thinsp;| < κ, and α < κ for all α ∈ A, then sup A < κ.

If κ is a strongly inaccessible cardinal, then transfinite induction proves |&thinsp;Vα&thinsp;| < κ for all α < κ:
 * 1) α = 0: |&thinsp;V0&thinsp;| = 0.
 * 2) For α+1: |&thinsp;Vα+1&thinsp;| = |&thinsp;Vα ∪ P(Vα)&thinsp;| ≤ |&thinsp;Vα&thinsp;| + 2 undefined = 2 undefined < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.
 * 3) For limit α: |&thinsp;Vα&thinsp;| = |&thinsp;∪ξ < α Vξ&thinsp;| ≤ sup {|&thinsp;Vξ&thinsp;| : ξ < α} < κ. Last inequality uses inductive hypothesis and κ being strongly inaccessible.

To prove Theorem 1: since a set X enters Vκ only through P(Vα) for some α < κ, we have X &sube; Vα. Since |&thinsp;Vα&thinsp;| < κ, we have |&thinsp;X&thinsp;| < κ. Conversely: if a class X has |&thinsp;X&thinsp;| < κ, let β = sup {rank(x): x ∈ X}. Since κ is strongly inaccessible, |&thinsp;X&thinsp;| < κ, and rank(x) < κ for all x ∈ X, we have β = sup {rank(x): x ∈ X} < κ. Also, rank(x) ≤ β for all x ∈ X implies X &sube; Vβ+1, so X ∈ Vβ+2 &sube; Vκ. Therefore, X ∈ Vκ.

To prove Theorem 2, we compute: |&thinsp;Vκ&thinsp;| = |&thinsp;∪α < κ Vα&thinsp;| ≤ sup {|&thinsp;Vα&thinsp;| : α < κ}. Let β be this supremum. Since each ordinal in the supremum is less than κ, we have β ≤ κ. Now β cannot be less than κ. If it were, there would be a cardinal λ such that β < λ < κ; for example, take λ = 2&thinsp;. Since λ &sube; Vλ and |&thinsp;Vλ&thinsp;| is in the supremum, we have λ ≤ |&thinsp;Vλ&thinsp;| ≤ β. This contradicts β < λ. Therefore, |&thinsp;Vκ&thinsp;| = β = κ.

It can be shown that the sets and classes of Vκ satisfy all the axioms of NBG.