Axiom of pairing
Any two objects have a pair.
Wikipedia / Wikimedia Commons
The axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory, introduced by Zermelo in 1908 as a special case of his axiom of elementary sets. It states that given any two objects A and B, there exists a set C whose members are exactly A and B.
- field
- Axiomatic set theory, logic, mathematics, computer science
- introduced_by
- Ernst Zermelo
- year_introduced
- 1908
- part_of
- Zermelo–Fraenkel set theory
- formal_statement
- ∀A ∀B ∃C ∀D [D ∈ C ⇔ (D = A ∨ D = B)]
Lore & Background
The axiom of pairing is generally considered uncontroversial and appears in just about any axiomatization of set theory. In the standard formulation of Zermelo–Fraenkel set theory, it follows from the axiom schema of replacement applied to any given set with two or more elements, and thus is sometimes omitted. The existence of a set with two elements, such as { {}, { {} } }, can be deduced either from the axiom of empty set and the axiom of power set or from the axiom of infinity.
Reader's Guide
The axiom of pairing allows the construction of singletons and ordered pairs. The ordered pair (a,b) is defined as {{a},{a,b}}, which satisfies the condition that (a,b) = (c,d) if and only if a = c and b = d. Ordered n-tuples can be defined recursively. Weaker forms of the axiom exist: one version replaces the biconditional with a conditional, requiring only that A and B are members of some set C, from which the full pair can be extracted using the axiom schema of separation. Another weaker axiom is the axiom of adjunction, which differs by using D ∈ A instead of D = A. Together with the axiom of empty set and the axiom of union, the axiom of pairing can be generalized to a schema stating that for any finite number of objects A1 through An, there is a set whose members are precisely those objects. This schema can replace the axioms of empty set and pairing, but not the axiom of union.
Did You Know?
- The axiom of pairing was introduced by Zermelo in 1908 as a special case of his axiom of elementary sets.
- The set {A,A} is abbreviated {A}, called the singleton containing A.
- The axiom of pairing allows the definition of ordered pairs as (a,b) = {{a},{a,b}}.
- In the standard formulation of Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement.
Frequently Asked Questions
Who is the Axiom of Pairing?
It's a foundational axiom in Zermelo–Fraenkel set theory, originally penned by Ernst Zermelo back in 1908 as a special case of his broader axiom of elementary sets. Think of it as the rule that guarantees every two objects can be bundled into a single set together.
What are the Axiom of Pairing's powers and role?
Given any two objects A and B, it asserts the existence of a set C whose members are exactly A and B — nothing more, nothing less. Formally: ∀A ∀B ∃C ∀D [D ∈ C ⇔ (D = A ∨ D = B)]. It's the minimal 'glue' that lets you form a two-element collection from arbitrary inputs.
How does the Axiom of Pairing's origin story go?
Zermelo introduced it in 1908 not as a standalone axiom but as a special case within his axiom of elementary sets. It was later carved out and adopted as its own axiom in the ZF framework, where it has remained ever since.
Why is the Axiom of Pairing so important to the bigger picture?
Without it you couldn't even construct a two-element set, which means ordered pairs, functions, relations, and virtually all of higher mathematics would have no footing. It's the smallest building block that makes the rest of set-theoretic construction possible.
Does the Axiom of Pairing only work when A and B are already sets?
No — the quantifiers range over arbitrary objects, so it applies even if one or both are urelements (non-set entities). This makes it slightly more general than the 'pairing of sets' phrasing sometimes used in informal discussions.
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