Ur-element

Definition

In set theory an ur-element or urelement is something which is not a set, but may itself be an element of a set. That is, if U is a ur-element, it makes no sense to say

X∈U,

although

U∈X

is perfectly legitimate.

This should not be confused with the empty set where saying

X∈<math>\varnothing<math>

is logically reasonable, but merely false.

Ur-elements are also sometimes known as "atoms" or "individuals."

Ur-elements and axiomatization

In the standard axiomatization of set theory known as Zermelo-Fraenkel set theory, there are no ur-elements. However, other axiomatizations do use ur-elements, see for example: Kripke-Platek set theory with urelements. In systems, such as set theory with types, a ur-element is sometimes an object of type 0, hence the name "atom." In such theories, the axiom of extensionality requires special formalization and treatment.