Axiom of Pairing

The Axiom of Pairing is a possible axiom in Material set theory: #m/def/set/zf

(\forall 𝑥) (\forall 𝑦) (\exists 𝔐 𝐴) [𝑧 \in 𝐴 ⟺ 𝑧 = 𝑥 \lor 𝑧 = 𝑦]

which is to say, for any two objects (possibly the same) there is a set whose only elements are those two objects. It follows from the Axiom of Extensionality that such a doubleton $𝐴$ is unique, which we denote ${𝑥, 𝑦}$ .

Axiom of Pairing for classes

In a material set theory with classes, we must modify the axiom slightly, since we cannot pair proper classes: #m/def/set/nbg

(\forall 𝔈 𝑥) (\forall 𝔈 𝑦) (\exists 𝔐 𝑧) [𝑧 \in 𝐴 ⟺ 𝑧 = 𝑥 \lor 𝑧 = 𝑦]

#state/develop | #lang/en | #SemBr