Russell's paradox

Etymology
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Proper noun

 * 1)  The paradox that a set defined to contain all sets which do not contain themselves can neither consistently contain itself nor not contain itself.
 * 2) * 2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, (Open Court), page 43,
 * Roughly, the idea is that Russell's paradox reveals that certain logics suffer serious problems, and therefore these logics should be avoided.Here again, Quine asserts that the real lesson of Russell's paradox is that we should give up quantifying over abstracta.
 * 1) * 2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, (Open Court), page 43,
 * Roughly, the idea is that Russell's paradox reveals that certain logics suffer serious problems, and therefore these logics should be avoided.Here again, Quine asserts that the real lesson of Russell's paradox is that we should give up quantifying over abstracta.

Usage notes
The paradox can be stated as follows:
 * Define $$\textstyle R=\{x:x\notin x\}$$.
 * Either (a) $$\textstyle R\in R$$ or (b) $$\textstyle R\notin R$$.
 * In case (a), $$\textstyle R\in R\Rightarrow R\notin R$$; in case (b), $$\textstyle R\notin R\Rightarrow R\in R$$.

In the standard axiomatisation of set theory (ZFC), the paradox is avoided by disallowing the definition of sets with criteria of .

Translations

 * Finnish: Russellin paradoksi
 * Icelandic: Russell-þversögn