Ur-element
Encyclopedia
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."
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.
This should not be confused with the empty set
where saying
is well-formed but false.
Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality
must be formulated to apply only to objects that are not urelements.
This situation is analogous to the treatments of theories of sets and classes
. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are minimal objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally.)
of 1908 included urelements. It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus standard expositions of the canonical axiomatic set theories ZF
and ZFC do not mention urelements. (For an exception, see Suppes.) 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. In type theory
, an object of type 0 can be called an urelement; hence the name "atom."
Adding urelements to the system New Foundations
(NF) to produce NFU has surprising consequences. In particular, Jensen 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
.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an urelement or ur-element (from the German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
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."
Theory
There are several different but essentially equivalent ways to treat urelements in a first-order theory.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.
This should not be confused with the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
where saying
is well-formed but false.
Another way is to work in a one-sorted theory with a unary relation used to distinguish sets and urelements. As non-empty sets contain members while urelements do not, the unary relation is only needed to distinguish the empty set from urelements. Note that in this case, the axiom of extensionality
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
must be formulated to apply only to objects that are not urelements.
This situation is analogous to the treatments of theories of sets and classes
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
. Indeed, urelements are in some sense dual to proper classes: urelements cannot have members whereas proper classes cannot be members. Put differently, urelements are minimal objects while proper classes are maximal objects by the membership relation (which, of course, is not an order relation, so this analogy is not to be taken literally.)
Urelements in set theory
The Zermelo set theoryZermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
of 1908 included urelements. It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements. Thus standard expositions of the canonical axiomatic set theories ZF
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
and ZFC do not mention urelements. (For an exception, see Suppes.) Axiomatizations of set theory that do invoke urelements include Kripke–Platek set theory with urelements
Kripke–Platek set theory with urelements
The Kripke–Platek set theory with urelements is an axiom system for set theory with urelements that is considerably weaker than the familiar system ZF.-Preliminaries:...
, and the variant of Von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
described by Mendelson. In type theory
Type theory
In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...
, an object of type 0 can be called an urelement; hence the name "atom."
Adding urelements to the system New Foundations
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
(NF) to produce NFU has surprising consequences. In particular, Jensen 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
Axiom of infinity
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...
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
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
.