Russell's Paradox

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Important

Every set theory that contains an unrestricted comprehension principle leads to contradictions

⇒ unrestricted comprehension principle: there is a set of all and only the objects that have that property

Counter example

Let $R$ be the set of all sets that are not members of themselves. If $R$ is not a member of itself, then its definition entails that it is a member of itself. Yet if it is a member of itself, then it is not a member of itself since it is the set of all sets that are not members of themselves. In symbols:

$$\text{Let }R=\{x|x\notin x\},\text{then }R\in R⟺R\notin R$$

which is a paradox.