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Definition. A set is an ordinal if it is a transitive set of transitive sets.

This is the simplest definition of an ordinal I have ever encountered, and I happen to like it a lot for this reason. However, it seems to have one downfall: I am having a lot of trouble showing that the class of ordinals is well-ordered without using the axiom of regularity (foundation). Do I need to use regularity to show this? Are there models of $ZFC-(\text{regularity})$ in which the transitive sets of transitive sets are not well-founded?

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If you remove the axiom of foundation from ZF, then you cannot exclude the existence of, for instance, sets satisfying $x = \{x\}$. Such a set would be transitive, and have only transitive sets as elements, so it would fall under your definition of an ordinal.

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  • $\begingroup$ Huh! This makes me wonder: Why would anyone define ``ordinal'' the way I have above? This can't possibly be a useful definition in ill-founded set theory. $\endgroup$ – fauxefox Oct 8 '18 at 23:54

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