Burali-Forti paradox

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• noun set theory The following paradox: Let L be the set of all ordinal numbers. This set is well-ordered, so it has an ordinal number γ. Since L is the set of all ordinal numbers, L contains γ. But γ is equal to the well-ordered set of all ordinal numbers β smaller than γ. Thus, L is order isomorphic to one of its proper subsets, a contradiction.