Home > Axiom of pairing

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.

In the formal language of the Zermelo-Frankel axioms, the axiom reads:

or in words:

Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

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:

Any two sets have a pair.

{A,A} is abbreviated {A}, called the singletonIn mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as is also a singleton: the only element is a set (which itself is however not a singleton). A set is a singleton if and only if i containing A. Note that a singleton is a special case of a pair.

The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatizationIn mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. Every mathematical theory is based on a set of axioms. Usually these axioms are not mentioned when a mathematical equati of set theory.

Generalisation

Together with the axiom of empty setSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Frankel axioms, the axiom reads:, the axiom of pairing can be generalised to the following statement:

that is:

Given any finiteIn mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,. n with n isin N . It is a theorem that a set is finite if and only if there exists no bijection between the set and any of its prop number of sets A₁ through A_n, there is a set C whose members are precisely A₁ through A_n.

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

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 numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this n. The case n = 0 is simply the axiom of empty set. The case n = 1 is the axiom of pairing with A = A₁ and B = A₁. The case n = 2 is the axiom of pairing with A = A₁ and B = A₂. The cases n > 2 can be proved using the axiom of pairing and the axiom of unionSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the e multiple times. For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A₁,A₂}, the singleton {A₃}, and then the pair A₁,A₂},{A₃}}. The axiom of union then produces the desired result, {A₁,A₂,A₃}.

Thus, one may use this as an axiom schemaIn symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. For this reason, an axiom schema is used. Formally, an axiom schema is a set (usually infinite) of well formed formulae, each of whi in the place of the axioms of empty set and pairing. Normally, however, one uses the axioms of empty set and pairing separately, and then proves this as a theorem schema. Note that adopting this as an axiom schema will not replace the axiom of union, which is still needed for other situations.

Does anybody know what this axiom/theorem schema is called?