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A set is an ordinal if it is transitive and well-ordered by $\in$. How do the transitive sets linearly ordered by $\in$ look like? Are they again just ordinals by the regularity axiom?

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Yes, you're right. More generally, the $\in$-ordering on any set is well-founded (= every subset has a (possibly non-unique) minimal element; equivalently assuming AC, no descending chains exist) by regularity, and the well-founded linear orders are exactly the well-orders.

Meanwhile, dropping transitivity results in a much weaker notion: for example, every singleton is linearly ordered by $\in$. Every set linearly ordered by $\in$ is "$\in$-isomorphic" to an ordinal, however.

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