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History

Russell discovered the paradox in May or June 1901. 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". In a 1902 letter, 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:

There is just one point where I have encountered a difficulty. You state (p. 17) 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 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:

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. 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, and proposed a solution that Russell would endorse in his Principles of Mathematics, but was later considered by some to be unsatisfactory. For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.

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") laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell gave to the set-theoretic antinomies could have persuaded them 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". Footnote 9 is where he stakes his claim:

1903, 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.

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.

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.