This contradiction is famously illustrated by the barber paradox, describing a town barber who shaves all those, and only those, who do not shave themselves.

Mathematicians developed the Zermelo-Fraenkel axioms to enforce strict rules on set formation, which logically prevented any set from containing itself as a member.

Russell shared the paradox just as the second volume went to print. Frege's rushed addendum famously admitted that the foundational logic of his life's work had collapsed.