Naive set theory suggests that any clearly stated property should determine a set of all objects satisfying that property. Russell's paradox shows that this unrestricted principle is inconsistent.

The paradox

Consider the supposed set

$$R=\lbrace x:x\notin x\rbrace,$$

containing every set that is not an element of itself.

Ask whether

$$R\in R.$$

If $R\in R$, then by its definition $R\notin R$. If $R\notin R$, then it satisfies the defining property and therefore $R\in R$.

Either answer produces a contradiction.

What failed

The problem is not ordinary membership or set operations. It is the assumption that every predicate can be used without restriction to form a set containing exactly the objects satisfying it.

Axiomatic set theory

Modern foundations such as Zermelo–Fraenkel set theory specify explicit axioms governing how sets can be constructed and related. Sets are produced through permitted operations rather than an unrestricted “set of all things satisfying any property” rule.

Russell's paradox therefore marks the boundary between useful informal set language and the formal foundations needed when set theory itself becomes the object of study.