Axiom of Intersection

In typical material set theory such as $Z F$ , it is not necessary to take the existence of an intersection as an axiom, as it can be derived from the Axiom of Union and Specification Axiom Schema. That is, it is a theorem of $Z F$ that #m/thm/set/zf

⊢ (\forall 𝔐 E) (\exists 𝔐 𝐵) [𝑥 \in 𝐵 ⟺ (\forall 𝔐 𝑋 \in E) [𝑥 \in 𝑋]]

which is to say, for any set $E$ there exists an intersection $𝐵$ consisting of the elements of elements of $E$ , which by extensionality is unique, and we denote $⋂ E$ .

Proof in

Z F

By the Axiom of Union, $⋃ E$ exists. Consider the formula

𝜙 (𝑥) ⟺ (\forall 𝔐 𝑋 \in 𝐸) [𝑥 \in 𝐸]

then using this with the Specification Axiom Schema on $⋃ E$ gives

𝐵 = {𝑥 \in ⋃ E : 𝜙 (𝑥)}

which is the $𝐵$ demanded above.

Axiom of Intersection for classes

For classes, on the other hand, the Axiom of Intersection¹ plays an imortant role in replacing the Specification Axiom Schema:

(\forall ℭ 𝔩 𝔰 𝑋) (\forall ℭ 𝔩 𝔰 𝑌) (\exists ℭ 𝔩 𝔰 𝑍) (\forall 𝑢) [𝑢 \in 𝑍 ⟺ 𝑢 \in 𝑋 \land 𝑢 \in 𝑌]

#state/tidy | #lang/en | #SemBr

2015. Introduction to Mathematical Logic, §4.1, p. 236 ↩

Axiom of Intersection

Axiom of Intersection for classes

Footnotes