Axiom of union
Axiom asserting the union of a set of sets is a set.
The axiom of union is one of the axioms of Zermelo–Fraenkel set theory, introduced by Ernst Zermelo. Informally, it states that if X is a set of sets, then the union of all sets in X is still a set.
- field
- Axiomatic set theory
- known_for
- Axiom of union in Zermelo–Fraenkel set theory
- introduced_by
- Ernst Zermelo
Lore & Background
The axiom of union is one of the axioms of Zermelo–Fraenkel set theory, introduced by Ernst Zermelo. Informally, it states that if X is a set of sets, then the union of all sets in X is still a set. In formal language, the axiom reads: ∀X ∃Y ∀u (u ∈ Y ↔ ∃z (u ∈ z ∧ z ∈ X)).
Reader's Guide
The axiom of union allows one to unpack a set of sets and create a flatter set. Together with the axiom of pairing, it implies that for any two sets A and B, their binary union A ∪ B is also a set. Together with the axiom schema of replacement, it implies that one can form the union of a family of sets indexed by a set. The axiom is often used to construct the limit of an infinite sequence of sets. 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 ℶ_ω. However, many results in ZF(C) remain valid even without the axiom of union.
Did You Know?
- The axiom of union was introduced by Ernst Zermelo.
- Together with the axiom of pairing, it implies that the binary union A ∪ B is a set.
- The axiom is independent from the rest of the ZFC axioms.
- It is the only ZFC axiom that asserts the existence of singular strong limit cardinals such as ℶ_ω.
Frequently Asked Questions
Who is Axiom of union?
The Axiom of Union is a core rule in Zermelo–Fraenkel set theory that guarantees the union of any set of sets is itself a set. It was introduced by Ernst Zermelo in his original axiomatization and has remained a permanent fixture of the ZF canon ever since.
What is Axiom of union's role in the ZF story?
It acts as the bridge that lets you gather every element from every member of a given set into one new, well-defined set. Without it, standard constructions like building the natural numbers or defining functions would lack any formal justification within the system.
How does Axiom of union's story end?
It doesn't really end — it stands as an unconditional, always-available rule in ZF and ZFC with no exceptions or conditional clauses. Its premise simply requires that you already have a set whose elements are sets, and it then asserts the union exists.
Why is Axiom of union important to fans of set theory?
It is one of the few axioms that actively creates a new set from existing ones rather than merely restricting what may be formed. This makes it indispensable for almost every non-trivial construction in mathematics that takes place inside ZF.
Who brought Axiom of union into the canon?
Ernst Zermelo introduced it in his 1908 axiomatization of set theory, and it has been a standard, non-optional pillar of ZF ever since. No later revision of the system has removed or weakened it.
More in Set Theory And Foundations 1-20
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record