In the formal language of the ZermeloFrankel axioms, the axiom reads:
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 extension 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:
The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.
Together with the axiom of empty set, the axiom of pairing can be generalised to the following statement:
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 = 0 is simply the axiom of empty set. The case n = 1 is the axiom of pairing with A = A_{1} and B = A_{1}. The case n = 2 is the axiom of pairing with A = A_{1} and B = A_{2}. 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 {A_{1},A_{2}}, the singleton {A_{3}}, and then the pair {{A_{1},A_{2}},{A_{3}}}. The axiom of union then produces the desired result, {A_{1},A_{2},A_{3}}.
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.
Does anybody know what this axiom/theorem schema is called?
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