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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). Any of the stages V and L leading to the construction of the von Neumann universe V and Gödel''s constructible universe L are transitive sets. The universes L and V themselves are transitive classes.