Imaginary unit

Template:No footnotes 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.

Formal statement

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

$\mathrm{\forall }A\mathrm{\forall }B\mathrm{\exists }C\mathrm{\forall }D[D\in C⟺(D=A\vee D=B)]$

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.

or in simpler words:

Given two sets, there is a set whose members are exactly the two given sets.

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:

Any two sets have a pair.

{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:

$(a,b)=\{\{a\},\{a,b\}\}.$

Note that this definition satisfies the condition

$(a,b)=(c,d)⟺a=c\wedge b=d.$

Ordered n-tuples can be defined recursively as follows:

$(a_1,\dots ,a_n)=((a_1,\dots ,a_{n-1}),a_n).$

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 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.

Generalisation

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

$\mathrm{\forall }{A}_1\dots \mathrm{\forall }{A}_n\mathrm{\exists }C\mathrm{\forall }D[D\in C⟺(D=A_1\vee \cdots \vee D=A_n)]$

that is:

Given any finite 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 number n.

• 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 union 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}. We can extend this schema to include n=0 if we interpret that case as the axiom of empty set.

Thus, one may use this as an axiom schema 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.

Another alternative

Another axiom which implies the axiom of pairing in the presence of the axiom of empty set is

$\mathrm{\forall }A\mathrm{\forall }B\mathrm{\exists }C\mathrm{\forall }D[D\in C⟺(D\in A\vee D=B)]$.

Using {} for A and x for B, we get {x} for C. Then use {x} for A and y for B, getting {x,y} for C. One may continue in this fashion to build up any finite set. And this could be used to generate all hereditarily finite sets without using the axiom of union.

References

• Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.