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...

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...

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...

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...

are no other points of S Uniqueness quantification – Logical quantifier Urelement – Concept in set theory Stoll, Robert (1961). Sets, Logic and Axiomatic...

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

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...

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...

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...

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

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)...

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...

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...

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...

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...

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

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

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...

Admissible ordinal Hereditarily countable set Kripke–Platek set theory with urelements Poizat, Bruno (2000). A course in model theory: an introduction to contemporary...

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

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...

Alternative set theory Axiomatic set theory Kripke–Platek set theory with urelements Morse–Kelley set theory Naive set theory New Foundations Positive set...

hereditary set is interesting only in a context in which there may be urelements. The inductive definition of hereditary sets presupposes that set membership...

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

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...

National Formosa University, a university in Taiwan New Foundations with Urelements, an axiomatic set theory in mathematical logic No first use, a nuclear...

(1967) are set theory texts built around MK; Rubin's ontology includes urelements. These authors and Mendelson (1997: 287) submit that MK does what is expected...