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Concept
Axiom of pairing
Summary
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets.
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
In words:
Given any object A and any object B, there is a set C such that, given any object D, D is a member of C if and only if D is equal to A or D is equal to B.
Or in simpler words:
Given two objects, there is a set whose members are exactly the two given objects.
As noted, what the axiom is saying is that, given two objects A and B, we can find a set C whose members are exactly A and B.
We can use the axiom of extensionality to show that this set C is unique.
We call the set C the pair of A and B, and denote it {A,B}.
Thus the essence of the axiom is:
Any two objects have a pair.
The set {A,A} is abbreviated {A}, called the singleton containing A.
Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the infinitely descending chains from the Axiom of regularity.
The axiom of pairing also allows for the definition of ordered pairs. For any objects and , the ordered pair is defined by the following:
Note that this definition satisfies the condition
Ordered n-tuples can be defined recursively as follows:
The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, and thus it is sometimes omitted. The existence of such 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.
In the absence of some of the stronger ZFC axioms, the axiom of pairing can still, without loss, be introduced in weaker forms.
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