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L ogic This paradox of the greatest ordinal was the first paradox discovered in modern set theory and was formulated by Cesare Burali-Forti. An ordinal number can be assigned to every well-ordered set, that is, a set for which every subset has at least one member. Such ordinals can be compared for size, and the set of these ordinals is a well-ordered set. The ordinal of this set must be larger than any ordinal contained within the set, but because the set is of all ordinals of well-ordered sets, the ordinal of the set must be contained within it. The ordinal of this set is therefore larger than and not larger than any ordinal within the set. According to Russell , the way of solving this paradox is to deny that the set of all ordinal numbers is well-ordered. “It is that in order to avert Burali-Forti's paradox the authors of Principia felt called upon to suspend typical ambiguity and introduce explicit type indices at the crucial point.” Quine, Selected Logical Papers ... log in or subscribe to read full text