Axiom of pairing

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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.

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Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:

or in words:

or in simpler words:

Interpretation

What the axiom is really saying is that, given two sets A and B, we can find a set C whose members are precisely 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:

{A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.

The axiom of pairing also allows for the definition of ordered pairs. For any sets a and b, the ordered pair is defined by the following:

Note that this definition satisfies the condition

Ordered n-tuples can be defined recursively as follows:

Non-independence

The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows 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.

Generalisation

Together with the axiom of empty set, the axiom of pairing can be generalised to the following schema:

that is:

This set C is again unique by the axiom of extension, and is denoted {A1,...,An}.

Of course, we can't refer to a finite number of sets rigorously without already having in our hands a (finite) set to which the sets in question belong. Thus, this is not a single statement but instead a schema, with a separate statement for each natural number n.