How does Russell’s paradox arise from unrestricted set formation?

Russell’s paradox arises from the use of an unrestricted comprehension principle in early set theory. This principle asserts that for any condition expressible by a formula $\varphi (x)$, one can form the set of all sets $x$ that satisfy that condition, denoted as $\{x\mid \varphi (x)\}$.1

The paradox emerges when this principle is applied to the condition of non-self-membership, defined by the formula $x\notin x$. This allows for the construction of a set $R$, defined as the set of all sets that are not members of themselves:
$R=\{x\mid x\notin x\}$12

The contradiction occurs when evaluating whether $R$ is a member of itself:
* If $R$ is a member of itself ($R\in R$), it must satisfy the condition for membership in $R$, which is $R\notin R$.
* If $R$ is not a member of itself ($R\notin R$), it meets the condition for membership in $R$, meaning $R\in R$.

This "vicious circle" demonstrates that an unfettered comprehension principle leads to logical inconsistency. Modern axiomatic set theory avoids this by restricting comprehension, ensuring that sets can only be formed as subsets of existing sets rather than from the entire universe of all sets.13