Full Idea
The terms 'set' and 'is a member of' are primitives of Zermelo's 1908 axiomatization of set theory. They are not given model-theoretic analyses or definitions.
Gist of Idea
Zermelo made 'set' and 'member' undefined axioms
Source
report of Ernst Zermelo (works [1920]) by Charles Chihara - A Structural Account of Mathematics 7.5
Book Reference
Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.204
A Reaction
This looks like good practice if you want to work with sets, but not so hot if you are interested in metaphysics.