Unordered pair

As a couple quantity, amount, or two pair is referred to in the set theory. Symbolized by set that contains exactly the objects and elements as Thus:

In the older, naive set theory, which was not axiomatized, the existence of a set described by extensional enumeration was intuitively justified. In axiomatic set teaching the existence of pair quantities required by a pair axiom since the Zermelo set theory of 1907, however. This axiom has been adopted in all major amount of lessons, for example in the Zermelo -Fraenkel set theory ZF or the Neumann - Bernays - Gödel set theory NBG. This pair axiom is verbal clarification: For all A and B, there is a set C that has exactly A and B as elements. In predicate logic specifying it is:

The pair axiom in ZF and NBG but a redundant axiom, because there can be derived from the other axioms as follows: Take the empty set by empty set axiom, twice is the power set by the power set axiom and thus obtain the special couple set whose elements by replacing axiom by any other elements may be replaced. In the older Zermelo - Fraenkel set theory without the axiom of replacement of 1921 this derivative was still impossible.

The required quantity in the pair axiom is clearly due to the Extensionalitätsaxioms and is listed in the form above. The type of elements the pair axiom says nothing. The objects may vary, depending on the chosen set theory. As part of ZF and NBG, both of which are a pure set theory, there are only quantities in a set theory with basic elements there may also be those, such as ZFUs.

An additional axiom for the singleton or singleton is not necessary. Indeed, the volume does not necessarily contain two different elements. In case there is only a singleton, since elements in quantities are not counted twice. Likewise, no axiom for larger quantities obtained by enumeration is needed because one gains larger finite sets successively over the union axiom. All of these quantities with extensional enumeration of the elements are so defined:

And so on.

Other significance

Sometimes the term " pair amount" used in the sense of a set of pairs for the Cartesian product of two sets.