# Why intersection is not an axiom in naive set theory by halmos

Why intersection is not an axiom in naive set theory? though, union was given as an axiom

$A\cap B=\{x\in A\mid x\in B\}$. So intersection can be deduced from comprehension axiom scheme.
The existence of intersection follows from the axiom schema of separation which informally tells you that if you start with a set and specify appropriately a way to select certain elements from this set by describing a formula that they satisfy then indeed there exists a subset containing precisely those elements. The construction of intersection fits this pattern (we can describe $A \cap B$ as selecting elements from $A$ that also lie in $B$) and so the existence of intersection follows from this axiom (for a rigorous proof, see Axiom schema of specification - Existence of intersection and set difference). The union construction generates from two sets $A,B$ a new set which is not a subset of either $A$ or $B$ in general and so requires another axiom to justify its existence.