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

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 but false.

This view of urelements is based on a two-sorted set theory, i.e., one having a domain containing two sorts of entities, namley sets and urelements. Alternatively, one may regard urelements as distinct empty sets in a one-sorted theory. In this case, the axiom of extensionality must be formulated and invoked with care.

The Zermelo set theory of 1908 included urelements. Subsequent research revealed that in the context of this and closely related axiomatic set theories, the urelements were of little mathematical value. Thus standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements. (For an exception, see Suppes 1960.) 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 in Mendelson (1997: 297-304). In type theory, an object of type 0 can be called a urelement; hence the name "atom."

Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen (1969) proved the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem. Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.^[1]

• Ronald Jensen (1969) "On the Consistency of a Slight(?) Modification of Quine's NF," Synthese 19: 250-63.
• Mendelson, Elliot (1997) Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall.
• Patrick Suppes (1960) Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972.

• Eric W. Weisstein, Urelement at MathWorld.