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∈
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.