Russell's paradox

Statement

There is no set $R$ such that, for each set $x$, $x \in R$ if and only if $x \notin x$.

Remarks

Proof: if such an $R$ exists, apply the condition to $x = R$. Then $R \in R$ if and only if $R \notin R$, which is a contradiction. So the collection $\{x \mid x \notin x\}$ is not a set. Unrestricted comprehension, which makes a set from each property, is inconsistent. It follows that there is no set of all sets. Otherwise, separation would give $R$ as a subset of it.

History

Bertrand Russell found the paradox in May 1901. Ernst Zermelo found it independently, by 1902 and possibly in 1899, but he did not publish it. In a 1902 letter, Russell told Gottlob Frege that the paradox arises in the logical system of Frege. Axiomatic set theories, such as Zermelo–Fraenkel set theory, avoid it with a restricted form of comprehension.

Explanations

L1 human

Try to form "the set of all sets that do not contain themselves". Ask whether this set contains itself: both answers lead to a contradiction. So some collections are too large to be sets.