material set theory

**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.) The standard example is ZFC.

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. A set’s identity here is determined by its elements, in other words the axiom of extensionality holds.

Relation to structural set theory is discussed in

- Michael Shulman,
*Comparing material and structural set theories*(arXiv:1808.05204)

See also

- Harvey Friedman (1997). Foundational Completeness. FOM, 1997 November.