Russell’s paradox is a famous paradox of set theory¹ that was observed around 1902 by Ernst Zermelo² and, independently, by the logician Bertrand Russell. The paradox received instantly wide attention as it lead to a contradiction in Frege’s monumental “Foundations of Arithmetic” (1893/1903) whose second volume was just about to go to print when Frege was informed about the inconsistency by Russell.

The paradox entangles a concept with its own extension in a vicious circle. The attempt to overcome this circularity in set formation had a huge impact on subsequent forms of axiomatic set theory and in the aftermath mathematical logic became heavily focussed on consistency proofs for fully specified formal theories: the paradoxes triggered a shift in the foundation of mathematics away from the mathematics to the foundation itself.

Doch zur Sache selbst! Herr Russell hat einen Widerspruch aufgefunden, der nun dargelegt werden mag.³

Statement

If one assumes a naive axiom of full comprehension, one can form the set

R={x|x∉x}. R = \{ x | x \notin x \}.

One then asks: is R∈RR\in R? If so, then R∉RR\notin R by definition, whereas if not, then R∈RR\in R by definition. Thus we have both R∈RR\in R and R∉RR\notin R, a contradiction.

Russell’s paradox is closely related to the classical liar paradox (“this sentence is false”), to Gödel’s incompleteness theorem, and to the halting problem — all use a diagonalization argument to produce an object which talks about itself in a contradictory or close-to-contradictory way.

On the other hand, Cantor's paradox can be said to “beta-reduce” to Russell’s paradox when we apply Cantor's theorem to the supposed set of all sets. See Cantor's paradox for explanation.

Also related:

Resolutions

There are a number of possible resolutions of Russell’s paradox.

Russell himself (1903,1908,1910) proposed the introduction of type theory as a solution e.g. in Principia Mathematica (1910) an intricate system of ramified types tracks the variables of propositional functions in order to prevent circular propositions. This is inspired by Poincaré‘s ideas on impredicativity and can be viewed as a radical generalisation of Frege’s ontological distinction between an argument as a satured object and a function or concept as an unsaturated object.
The “classical” solution, adopted in ZFC and thus by the mathematical “mainstream”, is to restrict the axiom of comprehension so as to disallow the formation of the set RR: one requires that the set being constructed be a subset of some already existing set. The restricted axiom is usually given a different name such as the axiom of separation.
Another solution is to distinguish between sets and proper classes (= collections that are “too big” to be sets) as e.g. in NBG “set” theory.⁴ Here we may write down the definition of RR, but from R∉RR \notin R we may conclude R∈RR \in R only if we already know that RR is a set; the xx in the definition must be a set. So we have no contradiction, but only a proof that RR is a proper class.

In the set theory called New Foundations, the axiom of comprehension is restricted in a rather different way, by requiring the set-defining formula to be “stratifiable”. Since the formula x∉xx\notin x is not stratifiable, the set RR cannot be formed. This related to Russell’s ideas on ramified types.
In most structural set theories, the featurelessness of the elements of the structural sets secures the consistency of set formation. If sets cannot be elements of other sets, then the “definition” of RR is just a type error. The same is true in other structural foundational systems such as (modern, non-Russellian) type theory. However, Russell’s paradox can be recreated in structural foundations with inconsistent universes by constructing pure sets within them.
Alternatively, one can change the underlying logic. Passing to constructive logic does not help: although there is a seeming appeal to excluded middle (either R∈RR\in R or R∉RR\notin R), without using excluded middle we can obtain that RR is both not in RR and not not in RR, which is also a contradiction. However, passing to linear logic (or even affine logic) does help: there is an unavoidable use of contraction in the paradox. There exist consistent linear set theories? with the full comprehension axiom, in which R∈RR\in R implies R∉RR\notin R and vice versa, but we can never get both R∈RR\in R and R∉RR\notin R at the same time to produce a paradox. (See the Ph.D. thesis Linear Set Theory of Masaru Shirahata, completed under Grigori Mints supervision, Stanford University, 1994.)
Finally, and perhaps most radically, one can decide to allow contradictions, choosing to use a paraconsistent logic. There exist nontrivial paraconsistent set theories with full comprehension in which the set RR exists, being both a member of itself and not a member of itself.

References

Zermelo’s observation which is mentioned in a footnote of his 1908 paper on the well-ordering theorem is analyzed in

B. Rang, W. Thomas, Zermelo’s Discovery of the “Russell Paradox” , Hist. Math. 9 no.1 (1981) pp.15-22. https://doi.org/10.1016/0315-0860(81)90002-1

together with a transcription of a note dated 16 April 1902 by Husserl describing Zermelo’s proof.

Russell indicated the contradiction leading to the inconsistency of G. Frege’s system of “Grundgesetze der Arithmetik” in a famous letter to the latter on June 16th 1902 which together with Frege’s reply is reprinted pp.124-128 in

J. van Heijenoort (ed.), From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931 , Harvard UP 1967.

A re-typed version is available from the blog post here.

For an account of Russell’s encounter with the paradox:

I. Grattan-Guinness, How Russell Discovered his Paradox , Hist. Math. 5 no.2 (1978) pp.127-137.

The first published account is presumably in the appendix of

G. Frege, Grundgesetze der Arithmetik II , Pohle Jena 1903.

Russell discusses the paradox extensively in chapter X of

B. Russell, The Principles of Mathematics , Cambridge UP 1903.

The influence of Poincaré‘s views shows in

B. Russell, Les paradoxes de la logique , Revue de métaphysique et de morale 14 no.5 (1906) pp.627-650. (gallica)