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)...

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)...

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...

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

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

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...

Zermelo (1930). 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...

In axiomatic set theory and the branches of logic, mathematics, and computer science which rely upon it, the axiom of extensionality is one of the axioms...

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...

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...

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

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...

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

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

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

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...

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...

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...

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...

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

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

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...

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...

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...

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...

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

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...

Axiomatic set theory General set theory Kripke–Platek set theory with urelements Morse–Kelley set theory Naive set theory New Foundations Pocket set theory...

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