Editing Axiom of pairing (section)

=== Weaker ===
In the presence of standard forms of the [[axiom schema of separation]] we can replace the axiom of pairing by its weaker version:
:<math>\forall A\forall B\exists C\forall D((D=A\lor D=B)\Rightarrow D\in C)</math>.

This weak axiom of pairing implies that any given objects <math>A</math> and <math>B</math> are members of some set <math>C</math>. Using the axiom schema of separation we can construct the set whose members are exactly <math>A</math> and <math>B</math>.

Another axiom which implies the axiom of pairing in the presence of the [[axiom of empty set]] is the [[axiom of adjunction]]
:<math>\forall A \, \forall B \, \exists C \, \forall D \, [D \in C \iff (D \in A \lor D = B)]</math>.
It differs from the standard one by use of <math>D \in A</math> instead of <math>D=A</math>. 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 set]]s without using the [[axiom of union]].