How did Russell’s paradox reveal problems with unrestricted set formation?

Russell’s paradox revealed that the "unrestricted comprehension principle"—the assumption that any condition can define a set—leads to logical contradictions.

In early set theory, this principle allowed for the formation of a set of all sets that do not contain themselves, denoted as $R=\{x\mid x\notin x\}$. This creates a "vicious circle": if $R$ is a member of itself, it must satisfy the condition of not being a member of itself; if it is not a member of itself, it satisfies the condition of being a member of itself.1

This paradox demonstrated that one cannot simultaneously assume the existence of a "universal set" (the totality of all sets) and allow an unfettered ability to construct sets from any arbitrary condition. To resolve this, modern systems like Zermelo-Fraenkel set theory impose strict limitations on set formation, ensuring that a set cannot be as large as the universe of all sets, thereby preventing the construction of paradoxical entities.1