# Urelement

In set theory, a branch of mathematics, an **urelement** or **ur-element** (from the German prefix *ur-*, 'primordial') is an object that is not a set (has no elements), but that may be an element of a set. It is also referred to as an **atom** or **individual**. Ur-elements are also not identical with the empty set.

## Theory

There are several different but essentially equivalent ways to treat urelements in a

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.

Another way is to work in a

This situation is analogous to the treatments of theories of sets and

## Urelements in set theory

The

^{[3]}). Axiomatizations of set theory that do invoke urelements include Kripke–Platek set theory with urelements and the variant of Von Neumann–Bernays–Gödel set theory described by Mendelson.

^{[4]}In type theory

Adding urelements to the system

^{[6]}In finitist set theory

## Quine atoms

An alternative approach to urelements is to consider them, instead of as a type of object other than sets, as a particular type of set. **Quine atoms** (named after Willard Van Orman Quine) are sets that only contain themselves, that is, sets that satisfy the formula *x* = {*x*}.^{[7]}

Quine atoms cannot exist in systems of set theory that include the

^{[8]}

Quine atoms also appear in Quine's New Foundations, which allows more than one such set to exist.^{[9]}

Quine atoms are the only sets called **reflexive sets** by Peter Aczel,^{[8]} although other authors, e.g. Jon Barwise and Lawrence Moss, use the latter term to denote the larger class of sets with the property *x* ∈ *x*.^{[10]}

## References

**^**Dexter Chua et al.: ZFA: Zermelo–Fraenkel set theory with atoms, on: ncatlab.org: nLab, revised on July 16, 2016.- ISBN 0486466248.
**. Retrieved 17 September 2012.****. Retrieved 17 September 2012.****.****^**Holmes, Randall, 1998.*Elementary Set Theory with a Universal Set*. Academia-Bruylant.- ISBN 978-0-521-53361-4.
- ^ , retrieved 2016-10-17.
**.****.**

**
**## External links