basic constructions:
strong axioms
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
Russell’s paradox is a paradox of naive material set theory that was first observed by the logician Bertrand Russell. If one assumes a naive, full axiom of comprehension, one can form the set
One then asks: is $R\in R$? If so, then $R\notin R$ by definition, whereas if not, then $R\in R$ by definition. Thus we have both $R\in R$ and $R\notin R$, a contradiction.
Russell’s paradox is closely related to the liar paradox (“this sentence is false”), to Gödel’s incompleteness theorem, and to the halting problem — all use diagonalization? 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:
There are a number of possible resolutions of Russell’s paradox.
In Principia Mathematica Russell himself introduced a concept of “”ramified types (maybe the earliest type theory) in order to rule out the paradoxical operations.
The “classical” solution, adopted in ZFC and thus by most mainstream mathematicians, is to restrict the axiom of comprehension so as to disallow the formation of the set $R$: 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.
Essentially the same resolution is used in class theories such as NBG. Here we may write down the definition of $R$, but from $R \notin R$ we may conclude $R \in R$ only if we already know that $R$ is a set; the $x$ in the definition must be a set. So we have no contradiction, but only a proof that $R$ 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 $x\notin x$ is not stratifiable, the set $R$ cannot be formed. A similar (but more complicated) resolution was developed by Russell himself in his theory of ramified types.
In most structural set theories, there is no need to artificially restrict the set-formation rules: if sets cannot be elements of other sets, then the “definition” of $R$ 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 $R\in R$ or $R\notin R$), without using excluded middle we can obtain that $R$ is both not in $R$ and not not in $R$, 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 $R\in R$ implies $R\notin R$ and vice versa, but we can never get both $R\in R$ and $R\notin R$ at the same time to produce a paradox.
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 $R$ exists, being both a member of itself and not a member of itself.
Discussion of a paradox similar to Russell’s in type theory with W-types is in