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For a now slightly dated history of Hahn’s Theorem, see the my:

For a now slightly dated history of Hahn’s Theorem, see my:

Asaf: There are numerous proofs of Hahn’s Embedding Theorem in the literature besides the especially simple one due to Clifford. One proof is on pp. 56-60 of Laszlo Fuchs’s Partially ordered algebraic systems [1963] Pergamon Press. On page 60 of the just-said work there are also references to several other proofs including those of Clifford, Banaschewski, Gravett, Ribenboim and Conrad. Another proof, closely related to the one in Fuchs (including all preliminaries) can be found in Chapter 1 of Norman Alling’s Foundations of Analysis over Surreal Number Fields, North-Holland, 1987. Another very nice treatment, including all preliminaries, can be found in Chapter 1 of H. Garth Dales and W. H. Woodin’s Super-Real Fields, Oxford, 1996.

Finally, I note that the earliest, but largely forgotten, altogether modern proof of Hahn’s great theorem may be found on pp. 194-207 of Felix Hausdorff’s, Grundzüge der Mengenlehre, Leipzig [1914]. It was the lack of familiarity with Hausdorff’s proof, and the need for a concise, modern proof, that led to the plethora of proofs in the 1950s.

Asaf: There are numerous proofs of Hahn’s Embedding Theorem in the literature besides the especially simple one due to Clifford. One proof is on pp. 56-60 of Laszlo Fuchs’s Partially ordered algebraic systems [1963] Pergamon Press. On page 60 of the just-said work there are also references to several other proofs including those of Clifford, Banaschewski, Gravett, Ribenboim and Conrad. Another proof, closely related to the one in Fuchs (including all preliminaries) can be found in Chapter 1 of Norman Alling’s Foundations of Analysis over Surreal Number Fields, North-Holland, 1987. Another very nice treatment, including all preliminaries, can be found in Chapter 1 of H. Garth Dales and W. H. Woodin’s Super-Real Fields, Oxford, 1996.There is also an interesting proof in Jean Esterle's Remarques sur les théorèmes d'immersion de Hahn et Hausdorff et sur les corps de séries formelles, Quarterly Journal of Mathematics 51 (2000), pp. 2011-2019.

Finally, I note that the earliest, but largely forgotten, altogether modern proof of Hahn’s theorem may be found on pp. 194-207 of Felix Hausdorff’s, Grundzüge der Mengenlehre, Leipzig [1914]. It was the lack of familiarity with Hausdorff’s proof and the need for a concise modern proof that led to the plethora of proofs in the 1950s.

Amendment (Response to request for references)

Asaf: There are numerous proofs of Hahn’s Embedding Theorem in the literature besides the especially simple one due to Clifford. One proof is on pp. 56-60 of Laszlo Fuchs’s Partially ordered algebraic systems [1963] Pergamon Press. On page 60 of the just-said work there are also references to several other proofs including those of Clifford, Banaschewski, Gravett, Ribenboim and Conrad. Another proof, closely related to the one in Fuchs (including all preliminaries) can be found in Chapter 1 of Norman Alling’s Foundations of Analysis over Surreal Number Fields, North-Holland, 1987. Another very nice treatment, including all preliminaries, can be found in Chapter 1 of H. Garth Dales and W. H. Woodin’s Super-Real Fields, Oxford, 1996.

For a now slightly dated history of Hahn’s Theorem, see the my:

Hahn’s Über die nichtarchimedischen Grössensysteme and the Origins of the Modern Theory of Magnitudes and Numbers to Measure Them, in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic Publishers, 1995, pp. 165-213. (A typed version of the paper can be downloaded from my website: http://www.ohio.edu/people/ehrlich/)

Finally, I note that the earliest, but largely forgotten, altogether modern proof of Hahn’s great theorem may be found on pp. 194-207 of Felix Hausdorff’s, Grundzüge der Mengenlehre, Leipzig [1914]. It was the lack of familiarity with Hausdorff’s proof, and the need for a concise, modern proof, that led to the plethora of proofs in the 1950s.