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:
“Though your views are in straight antagonism to theirs, assume an identity of sentiment, assume that you are saying precisely that which all think, and in the flow of wit and love roll out your paradoxes in solid column, with not the infirmity of a doubt.”
—Ralph Waldo Emerson (1803–1882)
“The paradoxes of today are the prejudices of tomorrow, since the most benighted and the most deplorable prejudices have had their moment of novelty when fashion lent them its fragile grace.”
—Marcel Proust (1871–1922)
“The so-called paradoxes of an author, to which a reader takes exception, often exist not in the author’s book at all, but rather in the reader’s head.”
—Friedrich Nietzsche (1844–1900)