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.[1]
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
[edit]In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:[2] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , there is a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcup X} which consists of just the elements of the elements of that set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .
Consequences
[edit]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.[3]
Weaker form
[edit]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[4] states the axiom as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
[edit]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[lower-alpha 1] forms a model of Template:Tmath (the ZFC axioms minus the axiom of union).[3]
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A \cup B| \le |A| + |B| \le 2 \cdot |B| \le |B| \cdot |B| = |B \times B| \le |\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.[3]
Relation to Intersection
[edit]There is no corresponding axiom of intersection. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a nonempty set containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} , it is possible to form the intersection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap A} using the axiom schema of specification as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
[edit]- ↑ This condition is known as Template:Tmath being pseudo-hereditarily of cardinality less than Template:Tmath, to distinguish it from the stronger condition of being hereditarily of cardinality less than Template:Tmath, which requires the transitive closure itself to have cardinality less than Template:Tmath.
References
[edit]- ↑ Ernst Zermelo, 1908, "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65(2), pp. 261–281.
English translation: Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, pp. 199–215 ISBN 978-0-674-32449-7 - ↑ Jech, Thomas J. (1997). Set Theory (2nd ed.). Springer. p. 6. ISBN 978-3-540-63048-7.
- ↑ 3.0 3.1 3.2 Oman, Greg (2010). "On the axiom of union". Archive for Mathematical Logic. 49 (3): 283–289. doi:10.1007/s00153-009-0163-1. ISSN 0933-5846. Retrieved 2026-01-31.
- ↑ Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
Further reading
[edit]- 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.