Axiom of union
In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory, stating that, for any set x there is a set y whose elements are precisely the elements of the elements of x. It was introduced by Zermelo (1908). Together with the axiom of pairing this implies that for any two sets, there is a set (called their union) that contains exactly the elements of both. Together with the axiom of replacement the axiom of union implies that one can form the union of a family of sets indexed by a set.
Formal statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
- Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.
or, more simply:
- For any set , there is a set which consists of just the elements of the elements of that set.
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 the superset of the union of a set. For example, Kunen (1980) states the axiom as
which is equivalent to
Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions.
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 union of a set is a set.
Note that the union of A and B, commonly written as , can be written as . Thus, the ordinary construction of unions is trivial given the axiom of pairing.
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
- {c in E: for all D in A, c is 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.)
References
- 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.
- Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
- Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999. English translation: Heijenoort, Jean van (1967), "Investigations in the foundations of set theory", From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Source Books in the History of the Sciences, Harvard Univ. Press, pp. 199–215, ISBN 978-0-674-32449-7.
External links
- "Axiom of Union". PlanetMath.