The ordinals in set theory are well-ordered themselve, but do not form a set, but a class. But does there exists a set that has the same order structure as the ordinals?
2026-04-03 16:47:03.1775234823
A set with the same order structure as the ordinals
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No.
A set with the same structure as the ordinals would be well-ordered, and we know that every well-ordered set is isomorphic to an ordinal. In particular, not to the class of ordinals.
However, it is not hard to show that if you just want to talk about an elementary substructures, then there are many ordinals which are themselves elementary substructures of the class of ordinals. Even if we add up the arithmetical operators of ordinals. In that sense, it means, there are many ordinals which are themselves with the same properties as the class of ordinals.