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.

An angle is the union of two rays which have the same end point, but do not lie on the same line. If the angle is the union of $\overrightarrow {AB}$ and $\overrightarrow {AC}$, then these rays are called the sides of the angle; the [common] end point $A$ is called the vertex .

ZFC, 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 and precise formulation) that, taken together, define the core of mathematics through the usage of set theory.

2023年11月23日 · In terms of ZFC, this theory describes the properties of sets, and that include what sets can be constructed or otherwise proven to exist under the axioms of ZFC. It is, of course, possible to enumerate all sentences provable from ZFC.

2022年4月28日 · An 'F' angle is called a corresponding angle, a 'Z' angle is called an alternate angle and a 'C' angle is called a supplementary angle.

2020年7月1日 · 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 theory, regarded both as a field of mathematical research and as a foundation for ongoing mathematics (cf. also Axiomatic set theory).

2020年9月21日 · Bernays, anticipated by von Neumann, proposed a set theory with classes, $NBG$, adopted by Gödel in 1940, which was proved to be a conservative extension of $ZFC$. Quine, another influential advocate of first order logic, proposed New Foundations in 1937, later $NFU$, which also turned out to be bi-interpretable with $ZFC$.

2020年5月5日 · Perhaps one of the most significant advances in foundations is the identification of the consistency strength hierarchy. It allows us to calibrate mathematical statements with "canonical" extensions (guided by large cardinal axioms) or restrictions of ZFC.

ZFC is not a particular type of axiom, rather it refers to a collection of axioms in set theory. ZFC is Zermelo-Fraenkel (reference 1) along with the Axiom of Choice. Other axioms are at the foundation of other systems - for instance the 5 Euclidean axioms are what Euclidean geometry is …

以上八条公理合称 zf, 加入如下的选择公理 (ac) 后合称为 zfc. 公理 (选择) . 对任意不含空集的集合 S 存在一个选择函数 F 使得 F ( X ) ∈ X , ∀ X ∈ S .