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See all on WikipediaZermelo–Fraenkel set theory - Wikipedia
In 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 Russell's paradox, led to the desire for a more … 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 non-logical symbol, usually denoted 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 such as proper classes and the universal set.
Many mathematical theorems can be proven in much … See more1908Ernst 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 axiom set is from Kunen (1980). The … See more
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Proper classes (collections of mathematical objects defined by a property shared by … See moreWikipedia text under CC-BY-SA license ZFC | Brilliant Math & Science Wiki
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 …
ZFC - Encyclopedia of Mathematics
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 …
Axiom of infinity - Wikipedia
WEBAxiom of infinity. In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It …
Von Neumann universe - Wikipedia
WEBThis collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first …
Zermelo–Fraenkel set theory - Simple English Wikipedia, the free ...
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 …
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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 …
Axiom schema of replacement - Wikipedia
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 …
Zermelo-Fraenkel Set Theory (ZF) - Stanford Encyclopedia of …
WEBNote that the Replacement Schema can take you ‘out of’ the set \ (w\) when forming the set \ (v\). The elements of \ (v\) need not be elements of \ (w\). By contrast, the Separation …
Continuum hypothesis - Wikipedia
WEBIn Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: =, or even shorter with beth numbers: =. The …
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 …
Minimal model (set theory) - Wikipedia
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
WEBMay 24, 2024 · 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 …
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 。 …
set theory - Understanding ZFC - Mathematics Stack Exchange
WEBNov 23, 2023 · Understanding ZFC. Ask Question. Asked 6 months ago. Modified 6 months ago. Viewed 1k times. 8. I initially thought that ZFC was something like a collection of …
Axiom Of Choice | Brilliant Math & Science Wiki
WEBThe axiom of choice is an axiom in set theory with wide-reaching and sometimes counterintuitive consequences. It states that for any collection of sets, one can construct …
ツェルメロ＝フレンケル集合論 - Wikipedia
WEB外部リンク. ツェルメロ＝フレンケル集合論. 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 …
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） …
2024–25 Regionalliga - Wikipedia
WEBRegionalliga Nord. 18 teams from the states of Bremen, Hamburg, Lower Saxony and Schleswig-Holstein will compete in the 13th season of the reformed Regionalliga Nord. …
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Match Thread: 1st Match, Group A - United States of America vs …
WEB1 day ago · USA showing it is going to be a premier destination for world cricket. Great grounds in Texas, Miami, NY. Ground being built in Cali and Chicago. A dream tour for …
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