以下是来自于整理wikipedia关于罗素悖论的条目（https://en.wikipedia.org/wiki/Russell's_paradox）：

罗素在 1901年5月或6月发现了这个悖论。根据他自己在 1919 年的《数学哲学导论》中的描述，他“试图发现康托尔关于不存在最大基数的证明中的某些缺陷”。在1902年的一封信中，他向弗雷格宣布发现了弗雷格1879年 Begriffsschrift 中的悖论，并从逻辑和集合论的角度，特别是从弗雷格的函数定义的角度提出了这个问题：

只有一点我遇到了困难。您说（第 17 页 [上文第 23 页]），函数也可以充当不确定元素。这一点我以前是相信的，但现在我对这一观点产生了怀疑，因为存在以下矛盾。假设 w 是一个谓词：是一个不能谓之于自身的谓词。w能谓之自身吗？每一个答案都有得出其反面。因此，我们必须得出 w 不是谓词的结论。同样，也没有一个类（作为整体）是不属于自己的。由此我得出结论，在某些情况下，一个可定义的集合[Menge]并不构成一个总体。

罗素在 1903 年出版的《数学原理》中详细论述了这一问题，并在其中重复了他第一次遇到的悖论：

在结束对基本问题的讨论之前，有必要更详细地研究一下前面提到的关于谓词不能称谓自身的奇特矛盾。... 我可以说，我是在努力调和康托尔的证明中得到的...."

罗素就这一悖论写信给弗雷格，当时弗雷格正在准备他的《算术基础》第二卷。弗雷格很快给罗素做了回复；他 1902 年 6 月 22 日的信与 van Heijenoort 的评论一起发表在 Heijenoort 1967:126-127 中。弗雷格随后写了一个附录，承认了这一悖论，并提出了一个罗素将在其《数学原理》中认可的解决方案，但后来有人认为这一解决方案并不令人满意。

恩斯特-泽梅洛（Ernst Zermelo）在他（1908 年）的《关于有序排列可能性的新证明》（与他出版的“第一部公理集合论”同时出版）[21] 中声称，他事先发现了康托尔的朴素集合论中的二律背反。他说“然而，即使是罗素给出的集合论二律背反的基本形式也能说服他们[J. König, Jourdain, F. Bernstein]，这些难题的解决不在于有序（ordering），而只在于对集合概念的适当限制”：

91903，第 366-368 页。然而，我本人在罗素之外发现了这一二律背反（antinomy），并在 1903 年之前将其传达给了希尔伯特教授等人[23]。

弗雷格给希尔伯特寄去了他的《算术基础》（Grundgesetze der Arithmetik）；如上所述，弗雷格的最后一卷提到了罗素向弗雷格传达的悖论。在收到弗雷格的最后一卷后，希尔伯特于 1903 年 11 月 7 日给弗雷格写了一封信，信中提到罗素的悖论时说：“我相信泽梅洛博士三四年前就发现了它。在埃德蒙-胡塞尔（Edmund Husserl）的《笔记》（Nachlass）中发现了泽梅洛实际论证的书面记录”。

1923 年，路德维希-维特根斯坦提出对罗素悖论作如下“处理”：

一个函数之所以不能成为它自己的参数，是因为函数的符号已经包含了它的参数的原型，而它不能包含它自己。因为让我们假设函数 F(fx) 可以是它自己的参数：在这种情况下，就会有一个命题 F(F(fx)), 在这个命题中，外层函数 F 和内层函数 F 的含义一定不同，因为内层函数 F 的形式是 O(fx), 而外层函数 F 的形式是 Y(O(fx))。这两个函数中只有字母 "F "是共用的，而字母本身却没有任何意义。如果我们不把 F(Fu) 写成 (do) ： F(Ou) . Ou = Fu。这就解决了罗素悖论。(逻辑哲学论》，3.333）

罗素和怀特海撰写了三卷本的《数学原理》，希望实现弗雷格未能实现的目标。他们试图通过使用他们为此目的而设计的类型理论来消除朴素的集合论的悖论。虽然他们在某种程度上成功地为算术奠定了基础，但并不明显是通过纯粹的逻辑手段实现的。尽管《数学原理》避免了已知的悖论，并允许推导出大量的数学知识，但其体系也引发了新的问题。

无论如何，哥德尔（Kurt Gödel）在 1930-31年证明，虽然《数学原理》中的许多逻辑（现在被称为一阶逻辑）是完备的，但如果皮亚诺算术是一致的，它必然是不完备的。这被广泛地--尽管不是普遍地--认为证明了弗雷格的逻辑学计划是不可能完成的。

2001 年，在慕尼黑举行了庆祝罗素悖论一百周年的国际会议，会议论文集已经出版。

应用版本

这个悖论有一些更贴近现实生活的版本，对于非逻辑学家来说可能更容易理解。例如，“理发师悖论”假设有一个理发师，他给所有不刮胡子的人刮胡子，也只给不刮胡子的人刮胡子。当人们思考理发师是否应该给自己刮胡子时，悖论就开始出现了。

发师悖论等“外行版本”的简单反驳似乎是：不存在这样的理发师，或者理发师不是人，因此可以不存在悖论。罗素悖论的全部意义在于， “这样的集合不存在”的答案意味着给定理论中集合概念的定义不令人满意。请注意“这样的集合不存在”和“它是空集”这两种说法之间的区别。这就好比说“没有水桶”和说“水桶是空的”之间的区别。

二，wikipedia的原文

History

Russell discovered the paradox in May or June 1901.[12] By his own account in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal".[13] In a 1902 letter,[14] he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function:[a][b]

There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality.

Russell would go on to cover it at length in his 1903 The Principles of Mathematics, where he repeated his first encounter with the paradox:[15]

Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's proof…."

Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik.[16] Frege responded to Russell very quickly; his letter dated 22 June 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967:126–127. Frege then wrote an appendix admitting to the paradox,[17] and proposed a solution that Russell would endorse in his Principles of Mathematics,[18] but was later considered by some to be unsatisfactory.[19] For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.[20]

Ernst Zermelo in his (1908) A new proof of the possibility of a well-ordering (published at the same time he published "the first axiomatic set theory")[21] laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell9 gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set".[22] Footnote 9 is where he stakes his claim:

91903, pp. 366–368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.[23]

Frege sent a copy of his Grundgesetze der Arithmetik to Hilbert; as noted above, Frege's last volume mentioned the paradox that Russell had communicated to Frege. After receiving Frege's last volume, on 7 November 1903, Hilbert wrote a letter to Frege in which he said, referring to Russell's paradox, "I believe Dr. Zermelo discovered it three or four years ago". A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl.[24]

In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(fx) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of F(Fu) we write (do) : F(Ou) . Ou = Fu. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)

Russell and Alfred North Whitehead wrote their three-volume Principia Mathematica hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of naive set theory by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While Principia Mathematica avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.

In any event, Kurt Gödel in 1930–31 proved that while the logic of much of Principia Mathematica, now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely—though not universally—regarded as having shown the logicist program of Frege to be impossible to complete.

In 2001 A Centenary International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.[12]

Applied versions

There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the barber paradox supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.

An easy refutation of the "layman's versions" such as the barber paradox seems to be that no such barber exists, or that the barber is not a man, and so can exist without paradox. The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an empty set". It is like the difference between saying "There is no bucket" and saying "The bucket is empty ».