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- Zermelo–Fraenkel set theory
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See all on WikipediaIn set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with … See more
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as See more
Formally, ZFC is a one-sorted theory in first-order logic. Equality is treated as a primitive logical symbol and the signature has a single primitive … See more
One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann. In this viewpoint, the universe of set … See more
ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects … See more
1908Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory.1922Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity.1930Zermelo published a paper on the concept of limit numbers and sets of limit numbers.1978Abian and LaMacchia studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice.2003Jech published the third millennium edition of his book Set Theory.There are many equivalent formulations of the ZFC axioms; for a discussion of this, see Fraenkel, Bar-Hillel & Lévy 1973. The following particular … See more
Virtual classes
Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be … See moreWikipedia text under CC-BY-SA license WEBZFC, or Zermelo-Fraenkel set theory, is an axiomatic system used to formally define set theory (and thus mathematics in general). Specifically, ZFC is a collection of approximately 9 axioms (depending on convention …
WEBJul 1, 2020 · ZFC is the acronym for Zermelo–Fraenkel set theory with the axiom of choice, formulated in first-order logic. ZFC is the basic axiom system for modern (2000) set …
WEBVon Neumann universe. In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is …
WEBZermelo–Fraenkel set theory (abbreviated ZF) is a system of axioms used to describe set theory. When the axiom of choice is added to ZF, the system is called ZFC. It is the …
WEBThe continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, …
WEBZermelo-Fraenkel Set Theory (ZF) Axioms of ZF. Extensionality : \ (\forall x\forall y [\forall z (\left.z \in x\right. \leftrightarrow \left. z \in y\right.) \rightarrow x=y]\) This axiom asserts …
WEBIn 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 definable mapping is …
WEBIn set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC . The minimal model was introduced by Shepherdson ( 1951, 1952, 1953) and …
Axiom of global choice - Wikipedia
WEBThe axiom of global choice cannot be stated directly in the language of Zermelo–Fraenkel set theory (ZF) with the axiom of choice (AC), known as ZFC, as the choice function τ is …
Zermelo-Fraenkel Axioms -- from Wolfram MathWorld
WEB5 days ago · The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1), stands for exists, means for all, stands for "is an …
Large cardinal - Wikipedia
WEBA large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property. Most working set theorists believe …
Axiom of choice - Wikipedia
WEBIn mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is …
set theory - Understanding ZFC - Mathematics Stack Exchange
WEBNov 23, 2023 · First of all, ZFC is a collection of axioms, written in first-order logic. First order logic is defined in terms of primitive notions: constants, variables, functions, …
Axiom Of Choice | Brilliant Math & Science Wiki
WEBIntuitively, the axiom of choice guarantees the existence of mathematical objects which are obtained by a series of choices, so that it can be viewed as an extension of a finite …
ツェルメロ＝フレンケル集合論 - Wikipedia
WEB集合論 において、 ツェルメロ＝フレンケル集合論 （ 英: Zermelo-Fraenkel set theory ）とは、 ラッセルのパラドックス などのパラドックスのない 集合論 を定式化するため …
Axiomas de Zermelo-Fraenkel – Wikipédia, a enciclopédia livre
WEBNa matemática, a teoria dos conjuntos de Zermelo-Fraenkel com o axioma da escolha, nomeada em homenagem aos matemáticos Ernst Zermelo e Abraham Fraenkel e …
公理的集合論 - Wikipedia
WEB公理的集合論 （こうりてきしゅうごうろん、 axiomatic set theory ）とは、 公理化 された 集合論 のことである。 集合の公理系. ツェルメロ＝フレンケル集合論（ZF公理系） …
策梅洛-弗兰克尔集合论 - 维基百科，自由的百科全书
WEB策梅洛-弗兰克尔集合论 （英語： Zermelo-Fraenkel Set Theory ），是 数学基础 中最常用的 一階 公理化集合论 。 含 选择公理 時常简写为 ZFC ，不含選擇公理的則簡寫為 ZF …
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