Urelement

In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object (concrete or abstract) that is not a set, but that may be an element of a set. Urelements are sometimes called "atoms" or "individuals."

Theory

There are several different but essentially equivalent ways to treat urelements in a first-order theory.

One way is to work in a first-order theory with two sorts, sets and urelements, with a ∈ b only defined when b is a set. In this case, if U is an urelement, it makes no sense to say

although

is perfectly legitimate.

This should not be confused with the empty set where saying

is well-formed (albeit false) because ∅ is a set, whereas U is not.

Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements.