Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper *Über die nichtarchimedischen Grössensysteme* (Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse 116 (Abteilung IIa), 1907, pp. 601-655). Among the results established therein is Hahn’s Embedding Theorem, which is generally regarded to be the deepest result in the theory of ordered abelian groups. The following are two of its familiar formulations:

(i) Every ordered abelian group is isomorphic to a subgroup of a Hahn Group.

(ii) Every ordered abelian group G is isomorphic to a subgroup G’ of a Hahn Group, the latter of which is an Archimedean extension of G’.

(For definitions and modern proofs, see: A. H. Clifford [1954], *Note on Hahn’s theorem on ordered Abelian groups*, Proceedings of the American Mathematical Society, vol. 5, pp. 860–863; Laszlo Fuchs [1963], *Partially ordered algebraic systems*, Pergamon Press.)

Hahn’s proofs (and all subsequent proofs) of (i) and (ii) make use of the Axiom of Choice or some ZF-equivalent thereof. Moreover, while writing before the complete formulation of ZF (Foundation and Replacement had yet to be included), Hahn further maintained that he believed his embedding theorem could not be established without the well-ordering theorem, which had been established by Zermelo using Choice (and was subsequently shown to be equivalent in ZF to Choice). To my knowledge, this essentially amounts to the earliest conjecture that an algebraic result is equivalent (in ZF) to an assertion equivalent to the Axiom of Choice. Surprisingly, both Hahn’s use of Choice and his conjecture are overlooked in the well-known histories of the Axiom of Choice, including the excellent one by Gregory Moore. Apparently without knowledge of Hahn’s conjecture, D. Gluschankof (implicitly) asked if (i) is equivalent to the Axiom of Choice in ZF in his paper *The Hahn Representation Theorem for* ℓ-*Groups in ZFA*, (The Journal of Symbolic Logic, Vol. 65, No. 2 (Jun., 2000), pp. 519-52). However, Gluschankof did not answer the question and, unfortunately, died shortly after raising it. R. Downey and R. Solomon (in their paper *Reverse Mathematics, Archimedean Classes, and Hahn’s Theorem*) establish a countable version of Hahn’s theorem without using Choice, but their technique does not extend to the general case.

This leads to my two questions:

Has anyone established or refuted Hahn’s Conjecture?

Assuming (as I suspect) the answer to 1 is “no”, is the status of Hahn’s Conjecture the longest standing open question in Set Theory?

**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.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.

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

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