Suppose that for any coherent proposition $P(x)$, we can construct a set $\{x:P(x)\}$. Let $S=\{x:x\not\in x\}$. Suppose $S\in S$; then, by definition, $S\not\in S$. Likewise, if $S\not\in S$, then by definition $S\in S$. Therefore, we have a contradiction. Bertrand Russell gave this paradox as an example of how a purely intuitive set theory can be inconsistent. The regularity axiom, one of the Zermelo-Fraenkel axioms, was devised to avoid this paradox by prohibiting self-swallowing sets.
Remark. Russell’s paradox is the result of an axiom (due to Frege) in set theory, now obsolete, known as the axiom of (unrestricted) comprehension, which states: if $\phi$ is a predicate in the language of set theory, then there is a set that contains exactly those elements $x$ such that $\phi(x)$. In other words, $\{x\mid\phi(x)\}$ is a set. So if we take $\phi(x)$ to be $x\notin x$, we arrive at Russell’s paradox.