Axiom of union

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: \forall X\, \exists Y\, \forall u\, (u \in Y \leftrightarrow \exists z\, (u \in z \land z \in X)) or in words: :Given any set X, there is a set Y such that, for any element u, u is a member of Y if and only if there is a set z such that u is a member of z and z is a member of X. or, more simply: :For any set X, there is a set \bigcup X which consists of just the elements of the elements of that set X. == Consequences ==

The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the axiom of pairing, it implies that for any two sets and , their binary union is also a set. Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set. The axiom of union is often used to construct the limit of an infinite sequence of sets . For example, it can be used to construct the supremum of any set of ordinal numbers. == Weaker form ==

In the context of set theories which include the axiom schema of separation, the axiom of union is sometimes stated in a weaker form which only produces a superset of the union of a set. For example, Kunen states the axiom as \forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A] to facilitate its verification in various models. This form is logically equivalent to \forall \mathcal{F} \,\exists A \forall x [ [\exists Y (x \in Y \land Y \in \mathcal{F}) ] \Rightarrow x \in A]. Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions. == Independence ==

In its full generality, the axiom of union is independent from the rest of the ZFC-axioms. It is the only axiom that asserts the existence of singular strong limit cardinals such as (beth-omega, the limit of the sequence {{tmath|\aleph_0, 2^{\aleph_0}, 2^{2^{\aleph_0} }, \ldots}}). Concretely, if is a singular strong limit cardinal, then the set of sets such that and for all in the transitive closure of forms a model of {{tmath|\text{ZFC} - \text{Union} }} (the ZFC axioms minus the axiom of union). == Relation to Intersection ==

There is no corresponding axiom of intersection. If A is a nonempty set containing E, it is possible to form the intersection \bigcap A using the axiom schema of specification as \bigcap A = \{c\in E:\forall D(D\in A\Rightarrow c\in D)\}, so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as :{c: for all D in A, c is in D} is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a universal set is antithetical to Zermelo–Fraenkel set theory.) == Notes ==