This theorem is also known as the "Comparability theorem for well-orderings", https://proofwiki.org/wiki/Fundamental_Theorem_of_Well-Ordering.

It states that two well-ordered sets are either order isomorphic, or one is order isomorphic to an initial segment of the other. The proof is far from trivial and so I am surprised that it is difficult to find out who initially proved it (I haven't found it).

This article maybe hints it was Cantor: Page 10: "The second Beitragee developed the Grundlagen ideas by focusing on well-orderings and construing their order types as the ordinal numbers. Here at last was the general proof via order comparison of well-ordered sets that ordinal numbers are comparable."

Or maybe it's just one of those theorems that are too elementary to have a discoverer?

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    $\begingroup$ Cantor, G. (1897). Beiträge zur Begründung der transfiniten Mengenlehre. Math. Ann., 49(2), p. 215. (Transl. Jourdain) Cantor, G. (1915), p. 150. Contributions to the Founding of the Theory of Transfinite Numbers. is the location of the noted statement (statement N) and proof. See Ferreiros, J. (2008) Labyrinth of Thought. Springer. p. 235-236 regarding related ideas by Dedekind. $\endgroup$
    – Georg Essl
    Commented Jul 8 at 12:42


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