Axiom of the pair

In the formal Language of axiomatic of Zermelo-Fraenkel, the axiom is written:

$\backslash \; forall\; has\; \backslash \; \backslash \; forall\; B\; \backslash \; \backslash \; exists\; C\; \backslash \; \backslash \; forall\; X\; C\; \backslash \; Leftrightarrow\; (x=a\; \backslash \; vee\; x=b)$

who is read in French:

being given has and B two units, there exists a unit C such as, for any unit X , X is an element of C if and only if X is equal to has or with B .

The axiom expresses that, for two unspecified units has and B , it is possible to find a unit C whose elements are precisely has and B . The Axiome of extensionnality can be used to show that this unit C is single. The unit C is noted { has , B }. It is called Paire has and of B when has ≠ B , Singleton has , when has = B . In this last case { has , has } can be shortened in { has }.

In set theory, one considers sometimes that a singleton is a particular case of pair, for reasons of convenience of expression in the first developments. One thus speaks about the pair of has and of B even if it were not supposed that has ≠ B . It is contrary with the use in the remainder of mathematics, for example into combinative (when one counts the pairs of elements of a finished unit, one does not include/understand singletons them). Practically, the fields are sufficiently disjoined so that there is no ambiguity.

Axiom of pair is sufficiently simple and primitive, to seem axiom or to be demonstrable, under form possibly restricted (for example if the theory is typified), in any theory which axiomatizes the overall concept.

Generalization

$\backslash \; forall\; a\_1\; \backslash \; cdots\; \backslash \; forall\; a\_n\; \backslash \; \backslash \; exists\; C\; \backslash \; \backslash \; forall\; X\; C\; \backslash \; Leftrightarrow\; (x=a\_1\; \backslash \; vee\; \backslash \; cdots\; \backslash \; vee\; x=a\_n)$

who means that:

being given units has ₁,…, has _n it exists a unit C whose elements are precisely has ₁,…, has _n.

This unit C is still single according to the axiom of extensionnality, and is noted { has ₁,…, has _n}.

This generalization is well of a diagram of proposal: a proposal for each entirety thus an infinity of proposal. At this stage it is not necessary to have defined in set theory the concept of entirety, or overall finished. The entireties which intervene are necessarily those of the meta-language. A statement cannot have a number of quantifiers which depends on an object of the theory. For the entireties of the set theory it would be necessary to say the things differently. Each proposal of the diagram is thus associated with a natural entirety not no one N (of the meta-language). One can to add for N = 0, existence of empty set, which is of certain way particular case of diagram, if one remembers that the absurdity is, semantically, “neutral element” of disjunction:

∃c ∀x X ∉ C.

Each statement of the diagram is thus demonstrable in set theory. In any rigor one needs a recurrence in the meta-language to show that all these statements are theorems.

Diagram of replacement and axiom of the pair

The axiom of the pair could be omitted set theory of Zermelo-Fraenkel, because he results from the Schéma of axioms of replacement and from the Axiome of the whole of the parts. However one generally avoids doing it, because it intervenes, as of the first developments of the set theory, for example to define the couples, whereas the diagram of replacement is truly useful only for more advanced developments (ordinal for example). Here how it is deduced.

Are two unspecified units has and B , one wishes to show the existence of the pair { has , B } (possibly reduced to a singleton).

One first of all uses the existence of the empty set (demonstrable in set theory, starting from the Schéma of axioms of comprehension, therefore Schéma of axioms of replacement, to see Axiome of the empty set), noted ∅. According to the Axiome of the whole of the parts one can show the existence of the singleton {∅}, which is the whole of the parts of the empty set. One from of deduced, always by the same axiom, the existence of the pair {∅, {∅}} which is the whole of the parts of the singleton {∅}.

One now uses the functional relation in X and there following (see article Schéma of axioms of replacement):

( X = ∅ and there = has ) or ( X = {∅} and there = B )

This use of the diagram of replacement is rather not very characteristic of this last. It should be noted that at this moment of the development of the theory, one does not have the concepts of couple, of function…

See too

• axiomatic Set theory