[1][2] Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions.
In symbols: Russell also showed that a version of the paradox could be derived in the axiomatic system constructed by the German philosopher and mathematician Gottlob Frege, hence undermining Frege's attempt to reduce mathematics to logic and calling into question the logicist programme.
The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself.
[5] However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen.
Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proved true it destroys the conventional meaning of truth and falsity.
This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.
(Avoiding paradox was not Zermelo's original intention, but instead to document which assumptions he used in proving the well-ordering theorem.
In some extensions of ZFC, notably in von Neumann–Bernays–Gödel set theory, objects like R are called proper classes.
As José Ferreirós notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the replacement functions, can be 'entirely arbitrary' [ganz beliebig]"; the modern interpretation given to this statement is that Zermelo wanted to include higher-order quantification in order to avoid Skolem's paradox.
Around 1930, Zermelo also introduced (apparently independently of von Neumann), the axiom of foundation, thus—as Ferreirós observes—"by forbidding 'circular' and 'ungrounded' sets, it [ZFC] incorporated one of the crucial motivations of TT [type theory]—the principle of the types of arguments".
This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy.
(Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.)
The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of countable models (Skolem's paradox), but it enjoys some important advantages.
Through the work of Zermelo and others, especially John von Neumann, the structure of what some see as the "natural" objects described by ZFC eventually became clear: they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation.
It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics.
Yet another approach is to define multiple membership relation with appropriately modified comprehension scheme, as in the Double extension set theory.
[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".
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.
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.
[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".
A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl.
(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.
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 2001, A Centenary International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.
A notable exception to the above may be the Grelling–Nelson paradox, in which words and meaning are the elements of the scenario rather than people and hair-cutting.
One way that the paradox has been dramatised is as follows: Suppose that every public library has to compile a catalogue of all its books.
An example would be "paint": or "elect" In the Season 8 episode of The Big Bang Theory, "The Skywalker Intrusion", Sheldon Cooper analyzes the song "Play That Funky Music", concluding that the lyrics present a musical example of Russell's Paradox.