Material set theory

Axiom of Pairing

The Axiom of Pairing is a possible axiom in Material set theory: zf

(∀𝑥)(∀𝑦)(∃𝔐⁡𝐴)[𝑧∈𝐴⟺𝑧=𝑥∨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: nbg

(∀𝔈⁡𝑥)(∀𝔈⁡𝑦)(∃𝔐⁡𝑧)[𝑧∈𝐴⟺𝑧=𝑥∨𝑧=𝑦]

---

develop | en | SemBr