Russell's Paradox

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Russell's paradox is the most elementary contradiction that results from naive set theory. It is an immediate consequence of a schema of unrestricted comprehension (where all classes become sets). It, in part, justifies the exploration of axiomatic systems such as ZFC. It was first discovered by Bertrand Russell when reviewing Frege's "Die Grunderland der Arithmetik".

Statement of the paradox

Take the set of all sets that are not elements of themselves. Given the schema of unrestricted comprehension, any class is a member of this class if and only if it is not a member of itself. Therefore, this class is a member of itself if and only if it is not a member of itself, creating a contradiction.