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)...

due to Zermelo. Urelements are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other...

whenever x ∈ A {\displaystyle x\in A} , and x {\displaystyle x} is not an urelement, then x {\displaystyle x} is a subset of A {\displaystyle A} . Similarly...

Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set...

which are stronger than ZFC. The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets...

Logical property of being the one and only object satisfying a condition Urelement – Concept in set theory Stoll, Robert (1961). Sets, Logic and Axiomatic...

set theories. In set theories with urelements, one has to further make sure that the definition excludes urelements from appearing in ordinals. If α is...

early as 1922 that the axiom of choice may fail in a variant of ZF with urelements, through the technique of permutation models introduced by Abraham Fraenkel...

theory refer only to pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections...

axioms of Zermelo's set theory with urelements. Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice...

proposed in 1908 the inclusion of urelements, from which he constructed a transfinite recursive hierarchy in 1930. Such urelements are used extensively in model...

extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements Scott–Potter set theory Constructive set theory Zermelo set theory General...

Counter-examples to the reverse implications (from weak to strong) in ZF with urelements are found using model theory. Most of these finiteness definitions and...

consistency of NF. NF with urelements (NFU) is an important variant of NF due to Jensen and clarified by Holmes. Urelements are objects that are not sets...

theory) Atomic formula, a single predicate in first-order logic Atom, an urelement in set theory Intel Atom, a line of microprocessors Atom (system on chip)...

Choice. A modification of NF, NFU, due to R. B. Jensen and admitting urelements (entities that can be members of sets but that lack elements), turns out...

inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with urelements). This axiomatic system is useful to prove for example that every category...

of models in first-order logic. (See Löwenheim–Skolem theorem) urelement An urelement is something that is not a set but allowed to be an element of a...

Holden-Day. p. xix. ASIN B0006BQH7S. M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p...

of choice cannot be proved from the axioms of Zermelo set theory with urelements. 1931: Publication of Gödel's incompleteness theorems, showing that essential...

Administration of the German government Zermelo–Fraenkel set theory with atoms, a urelement This disambiguation page lists articles associated with the title ZFA...

urelements as well as sets, which makes it clear that there can be sets of entities defined by first-order theories not based on sets. The urelements...

formulas with bounded quantifiers, as in Kripke–Platek set theory with urelements. The axiom schema of specification is implied by the axiom schema of replacement...

Spezia Calcio owner Kripke–Platek set theory Kripke–Platek set theory with urelements This page lists people with the surname Płatek. If an internal link intending...

replacements: "If M is a set and each element of M is replaced by [a set or an urelement] then M turns into a set again" (parenthetical completion and translation...

John Henry Newman. Kuhn emphasizes history as the "Grundcharakter and Urelement" of Christianity. Yves Congar is considered the primary representative...

theory, called ZU because it is equivalent to Zermelo set theory with urelements Zu (fish), a genus of ribbonfish Ziauddin University Zeppelin University...

which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation...

all members of sets are themselves sets, but not in set theory with urelements. The axiom's usefulness can be seen from the fact that, if one accepts...

Korea Polytechnic University, South Korea Kripke–Platek set theory with urelements, an axiom system for set theory Kwantlen Polytechnic University, a public...