Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.
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The upper attic
The middle attic
The lower attic
The parlour
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The library
The cellar
Sources
Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
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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”.
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.