Why does set theory add the membership relation to logic?

In the context of the philosophy of logic and the foundations of mathematics, set theory incorporates the membership relation ($\in$) as a primitive, undefined term to provide a rigorous framework for mathematical discourse. By adding this relation to a formal system of logic—such as a modest form of the predicate calculus—logicians and mathematicians can define and construct mathematical objects, including numbers and functions, in terms of sets.123

The membership relation is essential for several reasons:

• Foundational Reduction: It allows the language of mathematics to be reduced to the language of set theory, where sets are collections of objects (members) and their properties are governed by specific axioms.23
• Defining Mathematical Structures: Through the membership relation and the axiom of extensionality, sets are uniquely determined by their members, enabling the formal definition of concepts like the null set, intersections, and unions.24
• Axiomatic Rigor: By treating membership as an undefined primitive, set theory avoids the inconsistencies of "naive" approaches (such as Russell’s paradox) by adopting axioms that restrict how sets can be formed, such as the axiom schema of separation.241