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[资料，科普，数学] 素数（99）：ZFC 的《斯坦福哲学百科全书》版本 Stanford Encyclopedia

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[资料，科普，数学] 素数（99）：ZFC 的《斯坦福哲学百科全书》版本 Stanford Encyclopedia

ZF： Zermelo–Fraenkel set theory，"the axiom of choice" is deleted

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

斯坦福哲学百科全书： Stanford Encyclopedia of Philosophy

ZF 系统：策梅洛-弗兰克尔（Zermelo-Fraenkel）集合论系统。

ZFC 系统：ZF，加上“选择公理”。

网传：ZFC公理集合论是万有理论，能够推导出经典数学的所有理论。

但是，公理集合论无法被证明是一致的，人们只是在事实上迄今为止未在其中发现悖论（矛盾）；并且，其中的选择公理的地位一直为人所质疑。虽然数学仍未建立在严格的基础之上，但20世纪30、40年代后，大部分数学家已不再关心数学基础的问题。

下面是 ZF (ZFC) 的《斯坦福哲学百科全书》 Stanford Encyclopedia of Philosophy版本：

一、ZFC 的 10条公理，《斯坦福哲学百科全书》 Stanford Encyclopedia of Philosophy

ZFC 的公理 The axioms of ZFC。

(1) Extensionality: 外延

If two sets A and B have the same elements, then they are equal.

(2) Null Set: 空集

There exists a set, denoted by ∅ and called the empty set, which has no elements.

(3) Pair: 配对

Given any sets A and B, there exists a set, denoted by {A,B}, which contains A and B as its only elements. In particular, there exists the set {A} which has A as its only element.

(4) Power Set: 幂集

For every set A there exists a set, denoted by P(A) and called the power set of A, whose elements are all the subsets of A.

(5) Union: 并集

For every set A, there exists a set, denoted by ⋃A and called the union of A, whose elements are all the elements of the elements of A.

(6) Infinity: 无穷

There exists an infinite set. In particular, there exists a set Z that contains ∅ and such that if A∈Z, then ⋃{A,{A}}∈Z.

(7) Separation: 分离

For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula φ of the first-order language of set theory.

Thus, Separation is not a single axiom but an axiom schema, that is, an infinite list of axioms, one for each formula φ.

(8) Replacement: 替换

For every given definable function with domain a set A, there is a set whose elements are all the values of the function.

Replacement is also an axiom schema, as definable functions are given by formulas.

(9) Foundation: 基础

Every non-empty set A contains an ∈-minimal element, that is, an element such that no element of A belongs to it.

These are the axioms of Zermelo-Fraenkel set theory, or ZF. The axioms of Null Set and Pair follow from the other ZF axioms, so they may be omitted. Also, Replacement implies Separation.

Finally, there is the Axiom of Choice (AC):

(10) Choice: 选择

For every set A of pairwise-disjoint non-empty sets, there exists a set that contains exactly one element from each set in A.

二、ZF 的（Axioms of ZF）的《斯坦福哲学百科全书》版本，英文全文的图片

Set Theory _ Zermelo-Fraenkel Set Theory (ZF) Stanford 130_裁剪_拉曲线.png