[资料，科普，数学] 素数（100）：NBG 的《大英百科全书》版本 Britannica

NBG： von Neumann-Bernays-Gödel Set Theory

大英百科全书： Encyclopedia Britannica, Britannica

ZFC： Zermelo–Fraenkel set theory with the axiom of choice

   冯·诺伊曼-博内斯-哥德尔集合论（NBG） von Neumann-Bernays-Gödel Set Theory，是一种与 ZFC 不同的集合论公理系统。

   下面是 NBG 的大英百科全书 Britannica 版本：

一、NBG 的 10条公理，《大英百科全书》 britannica

(1) Axiom of extension

   If A and B are classes and if, for all (sets) x, x ∈ A if and only if x ∈ B,then A = B.

(2) Axiom of the empty set

   There exists a set A such that, for all x, it is false that x ∈ A.

(3) Axiom schema for class formation

   If S(x) is a condition on x in which only set variables are introduced by the phrase "for all" or "for some" and in which B is not free, then there exists aclass B such that x ∈ B if and only if S(x).

(4) Axlom of pairing

   If A and B are sets, there exists a set (symbolized {A, B} and called the unordered pair of A and B) having A and B as its sole members.

(5) Axiom of union

   If C is a set, there exists a set A such that x ∈ A if and only if x ∈ B for some member B of C.

(6) Axiom of power set

   If A is a set, there exists a set B, called its power set, such that x ∈ B if and only if x ⊆ A.

(7) Axiom of infinity

   There exists a set A such that ∅ ∈ A and, if x ∈ A, then

(x ⋃{x})∈ A, in which x ⋃{x} is the set x with x adjoined as a further member.

(8) Axiom of choice

   If A is a set the elements of which are nonempty sets, then there exists a function f with domain A such that, for each member B of A, f(B) ∈ B.

(9) Axiom of replacement

   If (the class) X is a function and A is a set, then there exists a set B such that y ∈ B if and only if, for some x, (x,y) ∈X and x ∈A; i.e., the range of the restriction of a function X to a domain that is a set is also a set.

(10) Axlom of restriction (foundation axiom)

   Every nonempty class A contains an element B such that A∩B = ∅.

二、NBG 的《大英百科全书》版本，英文全文的图片

参考资料：

[1] 2022-07-13，策梅洛-弗兰克尔集合论/Zermelo-Fraenkel set theory/杜国平，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=1GmKrx&Type=bkzyb

[2] 2023-08-22，数学基础/foundations of mathematics/何浩平，中国大百科全书，第三版网络版[DB/OL]

https://www.zgbk.com/ecph/words?SiteID=1&ID=CI9Wg&Type=bkzyb

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