Material set theory (also called membership-based set theory) is a style of set theory that contrasts with structural set theory. In a material set theory, the elements of a set exist independently of that set. (The terminology ‘material’, or at least ‘materialistic’, goes back at least to Friedman 1997.)
Frequently in material set theory one takes everything to be a pure set, including the elements of sets themselves. Therefore, any two sets may be meaningfully compared to ask if they are equal or if one is a member of the other. As a slight variation (still material set theory), one may also accept ur-elements (or atoms) as elements. The main distinguishing feature of a material set theory is a global membership predicate, whereby it is meaningful to ask, given any object and a set, whether the object is an element of the set.