Axiom of Pairing - Generalisation

Generalisation

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

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.