Paradoxes
The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.
Read more about this topic: Class (set Theory)
Famous quotes containing the word paradoxes:
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—Walter Lippmann (1889–1974)
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—Friedrich Nietzsche (1844–1900)
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—Ralph Waldo Emerson (1803–1882)