Editing Axiom of union

{{short description|Concept in axiomatic set theory}}
{{Use shortened footnotes|date=May 2021}}  
In [[axiomatic set theory]], the '''axiom of union''' is one of the [[axiom]]s of [[Zermelo–Fraenkel set theory]]. This axiom was introduced by [[Ernst Zermelo]].{{r|Zermelo1908}}

Informally, the axiom states that for each set ''x'' there is a set ''y'' whose elements are precisely the elements of the elements of ''x''.

== Formal statement ==
In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:
:<math>\forall X\, \exists Y\, \forall u\, (u \in Y \leftrightarrow \exists z\, (u \in z \land z \in X))</math><ref>{{Cite book |last=Jech |first=Thomas J. |url=https://books.google.com/books?id=wmOISAAACAAJ |title=Set Theory |date=1997 |publisher=Springer |isbn=978-3-540-63048-7 |edition=2nd |pages=6 |language=en}}</ref>
or in words:
:[[Given any]] [[Set (mathematics)|set]] ''X'', [[Existential quantification|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'' [[logical conjunction|and]] ''z'' is a member of ''X''.
or, more simply:
:For any set <math>X</math>, there is a set <math>\bigcup X\ </math> which consists of just the elements of the elements of that set <math>X</math>.

== Relation to Pairing ==
The axiom of union allows one to unpack a set of sets and thus create a flatter set. 
Together with the [[axiom of pairing]], this implies that for any two sets, there is a set (called their [[union (set theory)|union]]) that contains exactly the elements of the two sets.

== Relation to Replacement ==
The axiom of replacement allows one to form many unions, such as the union of two sets.

However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms:{{cn|date=August 2019}} 
Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.

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.

== Relation to Separation ==
In the context of set theories which include the axiom 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{{r|Kunen1980}} states the axiom as 
: <math>\forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A].</math>
which is equivalent to
: <math>\forall \mathcal{F} \,\exists A \forall x [ [\exists Y (x \in Y \land Y \in \mathcal{F}) ] \Rightarrow x \in A].</math>
Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions.

== Relation to Intersection ==
There is no corresponding axiom of [[intersection (set theory)|intersection]]. If <math>A</math> is a ''nonempty'' set containing <math>E</math>, it is possible to form the intersection <math>\bigcap A</math> using the [[axiom schema of specification]] as
:<math>\bigcap A = \{c\in E:\forall D(D\in A\Rightarrow c\in D)\}</math>,
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.)

== References ==
{{reflist|refs=
<ref name=Zermelo1908>Ernst Zermelo, 1908, "Untersuchungen über die Grundlagen der Mengenlehre I", ''Mathematische Annalen'' 65(2), pp.&nbsp;261–281.<br>English translation: [[Jean van Heijenoort]], 1967, ''From Frege to Gödel: A Source Book in Mathematical Logic'', pp.&nbsp;199–215 {{isbn|978-0-674-32449-7}}</ref>

<ref name=Kunen1980>[[Kenneth Kunen|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier.  {{ISBN|0-444-86839-9}}.</ref>
}}

==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).
*[[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''.  Springer.  {{ISBN|3-540-44085-2}}.

{{Set theory}}
[[Category:Axioms of set theory]]

[[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]]