Axiom of union

Template:Use shortened footnotes In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo.

Informally, the axiom states that if Template:Tmath is a set of sets, then the union of all sets in Template:Tmath is still a set. In more basic terms, for each set Template:Tmath there is a set Template:Tmath whose elements are precisely the elements of the elements of Template:Tmath.

Formal statement

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 Template:Tmath and Template:Tmath, their binary union Template:Tmath 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 Template:Tmath. 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 Template:Tmath (beth-omega, the limit of the sequence Template:Tmath). Concretely, if Template:Tmath is a singular strong limit cardinal, then the set of sets Template:Tmath such that Template:Tmath and Template:Tmath for all Template:Tmath in the transitive closure of Template:Tmath forms a model of Template:Tmath (the ZFC axioms minus the axiom of union).

However, many results in ZF(C) remain valid even without the axiom of union. The axiom schema of replacement allows one to form a union if one can construct a set of the same or larger cardinality, together with a definable surjection from it onto that union, which can often be achieved with the axiom of power set. For example, the binary union Template:Tmath can be constructed when the cardinalities Template:Tmath and Template:Tmath are comparable, since without loss of generality one can assume that Template:Tmath and Template:Tmath, and then $$|A\cup B|\leq |A|+|B|\leq 2\cdot |B|\leq |B|\cdot |B|=|B\times B|\leq |{\mathcal {P}}({\mathcal {P}}(B))|$$ where the final inequality follows from the definition of the Cartesian product using the Kuratowski ordered pair.

Assuming the axiom of choice, the well-ordering theorem can be proved without the axiom of union, which can be used to show that Template:Tmath exists as long as the cardinalities of all sets in Template:Tmath are bounded. This condition is always true for finite unions, or more generally when Template:Tmath only contains sets with a finite number of infinite cardinalities. Conversely, assuming the consistency of ZF, there exists a model of Template:Tmath such that Template:Tmath never exists when Template:Tmath contains sets with an infinite number of infinite cardinalities.

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.)

Further reading

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.