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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:

  1. Has anyone established or refuted Hahn’s Conjecture?

  2. 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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A possibly related, and probably already answered, question: is "every linearly ordered set is well-ordered" equivalent to AC over ZF? The reason I ask is that it seems plausible for AC, restricted to linearly ordered sets, to be sufficient for Hahn's embedding theorem; and if that were the case, and the answer to the above were "no," then that would show that choice is strictly stronger than Hahn's theorem. –  Noah S Apr 27 '13 at 17:35
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@Noah. H. Rubin showed long ago that in ZF, Choice is equivalent to the assertion: Every set that can be ordered can also be well-ordered. –  Philip Ehrlich Apr 27 '13 at 18:16
    
Naoh. There are at least six or seven different proofs of Hahn's theorem in the literature and they tend to use AC or Zorn's Lemma or The Well-ordering Theorem in a variety of ways. In any case, I'm not sure I understand your question. –  Philip Ehrlich Apr 27 '13 at 20:31
    
@Philop: I'm sorry, I misread your response; I've deleted my comments following it. –  Noah S Apr 28 '13 at 6:29
    
Clifford's paper is starting by saying it is intended as a note for another paper, and he will not repeat the definitions; he also remarked another person proved something similar; Banaschewski's paper (which you have mentioned in the comments to my answer) is in German. Is there somewhere where all the relevant definitions and proofs are collected into a single file? If not, is there a few canonical references to read for the definitions besides the paper by Clifford? –  Asaf Karagila Apr 28 '13 at 14:45
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2 Answers

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).

  1. 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.

  2. 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:

Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375–381.

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).


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@Asaf: Many thanks on your informative answer to (2). I'd be very happy to hear about any subsequent thoughts you have on (1). Moreover, I'll take a look at the paper you cited. By the way, Banaschewski provided one of one the modern proofs of Hahn's Embedding Theorem. –  Philip Ehrlich Apr 27 '13 at 23:16
    
A conjecture that is very close to Schoenflies's conjecture, also still open (as far as I know), but apparently not asked in "ancient" times is that AC follows from "the ordering of cardinals is well-founded". Of course, in the presence of dependent choice this would be equivalent to Shoenflies's conjecture, but in ZF alone it looks strictly weaker. –  Andreas Blass Apr 28 '13 at 0:08
    
Andreas, yes this naturally comes up the moment you discuss Schoenflies' conjecture, although I'm not 100% sure that the two aren't equivalent in ZF. I wouldn't be that surprised by a proof that they are equivalent, or by a proof that they are not. –  Asaf Karagila Apr 28 '13 at 0:14
    
@Andreas, @Asaf, Do you know who should be credited with the well-foundedness conjecture? Or do you know of any references where it is mentioned? (I haven't found any, but I must surely be looking at the wrong places.) –  Andres Caicedo Apr 28 '13 at 1:58
    
@Andres, I don't recall seeing it anywhere either. It was a natural question that I have asked myself, and came up in most of my talks about it with my teachers in BGU (and perhaps once or twice in HUJI as well); most recently David Feldman's thread for open problems that you can explain a freshman was reopened (pun intended) and I posted both these conjectures, and Andreas posted pretty much the same comment. These are all the references I know, but there are no written ones that I have encountered in the literature. Strange... –  Asaf Karagila Apr 28 '13 at 2:33
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Regarding

  1. In 1902 Beppo Levi introduced the Partition Principle stating that if S is a partition of A then |S|≤|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.

This is a trivial thm of ZFC, but definitely not provable in ZF+DC, in particular, in the case A = the reals and S = the Vitali partition R/Q, as Sierpinski established in 1920s that any injection R/Q\to R implies a non-measurable set of reals, whose existence is not provable in ZF+DC by Solovay.

Regarding the oldest unsolved concrete problem in set theory, this is most likely one of Hausdorff's questions on pantachies, namely,

is there a pantachy containing no $(\omega_1,\omega_1^\ast)$ gap (H, Untersuchungen uber Ordnungstypen, V, 1907, May 05)

I underline that this is a concrete problem of existence of a certain mathematical object, rather than an abstract question on interrelations of different forms of AC spread over the whole set universe.

See more on this problem in my survey "Gaps in partially ordered sets" (12.1) in Hausdorff's Gesammelte Werke Band 1A, Springer 2013, 367-405, with additional references to Goedel (who rediscovered the problem with full ignorance of the Hausdorff's original formulation), Solovay, Kanamori.

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The Partition principle is in fact no provable in $\sf ZF+DC_\kappa$, for any $\kappa$. The reason is that we can easily engineer models in which $\sf ZF+\DC_\kappa$ holds, and there is a set which can be mapped onto $\kappa^+$, but has no subset of cardinality $\kappa^+$. –  Asaf Karagila May 21 '13 at 2:13
    
Also $\sf ZF+BPI$ can't prove the partition principle because it is consistent with $\sf ZF+BPI$ that there is an infinite Dedekind-finite set, which means that there is one which can be mapped onto $\omega$, and such since the set is Dedekind-finite there is injection from $\omega$ back to it. So all in all, it seems to be independent of the two "major" families of choice principles: restricted choices and the many relatives of $\sf BPI$. –  Asaf Karagila May 21 '13 at 2:35
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