Material set theory

Axiom of Intersection

In typical material set theory such as ZF, 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 ZF that zf

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

Proof in ZF

By the Axiom of Union, exists. Consider the formula

then using this with the Specification Axiom Schema on gives

which is the 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:

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Footnotes

1. 2015. Introduction to Mathematical Logic, §4.1, p. 236 ↩