relements

Relements is a term found in some set-theoretic literature used to describe urelements, i.e., objects that can be members of sets but are not themselves sets. The term is not standard; most contemporary texts prefer urelement or atom. In a theory that includes relements, the universe consists of two kinds of objects: urelements and pure sets. Urelements have no elements of their own, while sets may contain urelements or other sets. The membership relation is defined for sets, but urelements can occur as members within sets.

In extensions of ZF that admit urelements (often called ZFA or ZFU), Extensionality remains the principle for sets: two sets are equal if they have exactly the same members. Urelements participate in membership as atomic elements inside sets but are not themselves formed from previously existing elements. The Axioms of Foundations are typically applied to the portion of the universe that consists of sets, ensuring no infinite descending ∈-chains of sets; urelements do not introduce additional internal structure since they have no elements.

Models and construction: A basic model can be built by starting with a fixed nonempty set U of urelements and forming the cumulative hierarchy with U as the base. In such a framework, every object is either a urelement from U or a set whose elements are drawn from earlier stages. Within this setup, ordinary mathematics can be developed inside the set-theoretic universe, while urelements serve as primitive, indivisible symbols or objects.

See also: Urelement, Atom, ZFA, ZFU, Set theory with atoms.