In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of ZermeloFraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x. Together with the axiom of pairing this implies that for any two sets, there is a set that contains exactly the elements of both.
Contents
Formal statement
In the formal language of the ZermeloFraenkel axioms, the axiom reads:
or in words:
Interpretation
What the axiom is really saying is that, given a set A, we can find a set B whose members are precisely the members of the members of A. By the axiom of extensionality this set B is unique and it is called the union of A, and denoted . Thus the essence of the axiom is:
The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
Note that there is no corresponding axiom of intersection. If A is a nonempty set containing E, then we can form the intersection using the axiom schema of specification as
so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as
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 Zermeloâ€“Fraenkel set theory.)
References
 Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by SpringerVerlag, New York, 1974. ISBN 0387900926 (SpringerVerlag edition).
 Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3540440852.
 Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0444868399.
External links
Full article ▸
