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an actual contradiction

Russell’s paradox is an actual contradiction: if you assume you can quantify over all sets, then you derive a contradiction.

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a proof that the collection of all sets not containing themselves is a proper class

In the same way, Russell's paradox is a proof that the collection of all sets not containing themselves is a proper class.

a problem for our confidence that 2+2=4 is true

Is Russell's Paradox a problem for our confidence that 2+2=4 is true?

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a negated statement whose usual proof is a refutation by contradiction

Russell's paradox shows that every set theory that contains a ... , stated set-theoretically as 'there is no set whose elements are precisely those sets that do not contain themselves', is a negated statement whose usual proof is a refutation by contradiction.

the most famous of the logical or settheoretical paradoxes

Russell's paradox is the most famous of the logical or settheoretical paradoxes.

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the most famous of the set-theoretic paradoxes

Russell’s paradox is the most famous of the set-theoretic paradoxes; the set-theoretic paradoxesarises when one considers the set of all non self-membered sets, the Russell set.

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the most famous of the logical or set-theoretical paradoxes

Russell's paradox is the most famous of the logical or set-theoretical paradoxes.

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the theory that states: If you have a list of lists that do not list themselves, then that list must list itself, because that list doesn't contain itself

Russell's Paradox is the theory that states: If you have a list of lists that do not list themselves, then that list must list itself, because that list doesn't contain itself.

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which illustrates in a simple fashion the problems inherent in naive set theory

Russell discovers 'Russell's paradox' which illustrates in a simple fashion the problems inherent in naive set theory.

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an example of a flaw in set theory, which set theory

Not all problems are soluble without changing the system - Russell's Paradox (or the Barber of Baghdad) is an example of a flaw in set theory, which set theory can't readily be changed to fix.

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