Russell's paradox

English

Etymology

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Proper noun

1. Template:Lb The paradox that a set defined to contain all sets which do not contain themselves can neither consistently contain itself nor not contain itself.
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   • 2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, Template:W (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.Template:...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 $R=\{x:x\notin x\}$.
• Either (a) $R\in R$ or (b) $R\notin R$.
• In case (a), $R\in R\Rightarrow R\notin R$; in case (b), $R\notin R\Rightarrow R\in R$.

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

Synonyms

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Translations

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See also

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Further reading

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