The Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can...

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

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...

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

Well-founded set Well-order Power set Russell's paradox Set theory Alternative set theory Axiomatic set theory Kripke–Platek set theory with urelements Morse–Kelley...

set theory Pocket set theory Naive set theory S (set theory) Double extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements...

saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for...

pure sets and prevent its models from containing urelements (elements that are not themselves sets). Furthermore, proper classes (collections of mathematical...

Zermelo set theory sufficient for the Peano axioms and finite sets; Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and...

sets, called semisets. Even in systems related to ZFC, this scheme is sometimes restricted to formulas with bounded quantifiers, as in Kripke–Platek set...

Richard Platek, Kripke–Platek set theory Robert Platek, Spezia Calcio owner Kripke–Platek set theory Kripke–Platek set theory with urelements This page...

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

The power set of a transitive set without urelements is transitive. The transitive closure of a set X {\displaystyle X} is the smallest (with respect to...

quantification – Logical quantifier Urelement – Concept in set theory Stoll, Robert (1961). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp...

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

of choice is false in NF (without urelements). In 1969, Jensen showed that adding urelements to NF yields a theory (NFU) that is provably consistent....

Size. Monk (1980) and Rubin (1967) are set theory texts built around MK; Rubin's ontology includes urelements. These authors and Mendelson (1997: 287)...

cardinals. KP Kripke–Platek set theory Kripke 1.  Saul Kripke 2.  Kripke–Platek set theory consists roughly of the predicative parts of set theory Kuratowski...

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 set theories...

a transitive set of transitive sets. These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to further...

pure set theory, 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...

R(α) instead of Vα). If V0 is a set of urelements, the standard definition eliminates the urelements at V1. If X is a set, then there is a class Y such...

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

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

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any...

with Zermelo set theory without Choice. A modification of NF, NFU, due to R. B. Jensen and admitting urelements (entities that can be members of sets...

axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing...

from the 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...

In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly...

concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU...