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:
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Has anyone established or refuted Hahn’s Conjecture?
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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.
Best Answer
I don't know the answer to (1), and would be glad to give it some thought later this week. Regardless to (1) the answer to (2) is semi-negative.
There are two conjectures which seem to be slightly older (although not by much) than this of Hahn, although both were not explicitly stated as conjectures but since they went without proof, and sometimes there were disputes over the truth values of these statements -- so I prefer to think about them as conjectures (and in modern terms, I believe that would be right, too).
In 1905 Schoenflies asserted that the statement "There is no decreasing sequence of cardinals" implies the axiom of choice. This is still open, although in 1908 Zermelo rejected the claim.
In 1902 Beppo Levi introduced the Partition Principle stating that if $S$ is a partition of $A$ then $|S|\leq|A|$. Although he coined this principle to argue against its use by Bernstein, the latter rejected the criticism and claimed that this is one of the more important principles of set theory.
Of course all this was before Zermelo even introduced the axiom of choice in 1904. But in 1906 in an unpublished manuscript Russell claimed that AC is equivalent to PP, but this was without proof and the conjecture is still open to this very day.
More accurately, however, Russell proved that PP follows from another principle, and claimed the reverse implication holds as well (without proof), and in 1908 proved that the other principle is equivalent to the axiom of choice.
So we have two conjectures from 1906, regarding the equivalence of two statements to the axiom of choice. Neither has been proven yet, and there has been very little progress (to my knowledge) in obtaining any concrete answer. My opinion is that we lack the proper tools to handle the complex structure of cardinals in models without choice. But I digress.
All the information I gave here is taken from the following paper:
Along with the many hours that I have spent reading and searching results related to these topics (which made me rather certain that little progress has been made on these problems).