Ur-element

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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 an 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

$X \in \emptyset$

is logically reasonable, but merely false.

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

[edit] 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.

It has been found that adding urelements to the system NF to produce NFU has some surprising consequences. In particular, NFU is known to be consistent while the relative consistency of NF remains an open problem. In addition, NFU is consistent with the axiom of choice whereas NF disproves it.