Russell's paradox

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Russell's paradox

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Etymology

Named after English mathematician, logician and philosopher Bertrand Russell.

Proper noun

Russell's paradox

(set theory) The paradox that a set defined to contain all sets which do not contain themselves can neither consistently contain itself nor not contain itself.
- 1999, R. C. Penner, Discrete Mathematics: Proof Techniques and Mathematical Structures, World Scientific, page 109:
  One concludes that there must be something fishy about the Axiom of Comprehension, and, over time, the replacement of the Axiom of Comprehension by the Schema of Separation was seen to resolve Russell's paradox. Indeed, one cannot apply the Schema of Separation as in Russell's paradox unless one knows in advance that the collection of all sets is itself a set.
- 2001, M. Randall Holmes, “Tarski's Theorem and NFU”, in C. Anthony Anderson, Michael Zelëny, editors, Logic, Meaning and Computation: Essays in Memory of Alonzo Church, Springer (Kluwer Academic), page 469:
  The well-known theorem of Tarski that truth of sentences in any reasonably expressive language L cannot be defined in the language L itself is proved by a diagonalization argument similar to the argument involved in Russell's paradox. […] It is usual to think that Russell's paradox excludes "large" sets like the universe, but this is actually not the case. An alternate solution to Russell's paradox (and other paradoxes) was proposed by Quine (1937) in his system "New Foundations" (NF): comprehension restricted to stratified formulae.
- 2013, Greg Frost-Arnold, Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science, Carus Publishing Company (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 unrestricted comprehension.

Synonyms

(paradox in set theory): Russell's antinomy

Translations

paradox in set theory

Finnish: Russellin paradoksi
Icelandic: Russell-þversögn f