Philip Ehrlich
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory As I mentioned, Alling is very good regarding Clifford's proof plus it has a detailed discussion of all the relevant definitions. The treatment of Dales and Woodin is also quite good and self-contained, but quite different. If you read French, the treatment by Esterle is also very nice and quite short. All of these references are in the amendment to my question. |
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @Asaf: It would be helpful to have the paper of Hausner and Wendel as well as the aforementioned treatment of Alling available when looking at Clifford. |
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory No problem, Asaf. I agree with you, in that approach it is difficult to see if it is possible to overcome the reliance on a ZF-equivalent to choice. The other proofs tend to ware choice or Zorn or the well-ordering theorem on their sleeves. |
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory Good points, Joel. But if we take my question to be about ZFC, as it was intended, then that eliminates the status of CH as being an open question--doesn't it? |
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Apr 29 |
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @Asaf: If you can't read Banaschewski's proof in German, look at Paul Conrad's "A note on valued linear spaces", Proc. Amer. Math. Soc. 9 1958 646–647, which contains a slight generalization of the proof. There is also a good outline of Banaschewski's proof by Conrad in the latter's review of Banaschewski's paper for Math Rev. |
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Apr 29 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @Asaf: I don't have Gravett in front of me, but Clifford uses an equivalent of Choice more than you suggest. In the portion of the proof he's says mirrors Hausner and Wendel's proof for ordered vector spaces over the reals, explicit use is made of Zorn's Lemma. |
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Apr 28 |
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Apr 28 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @Asaf: For an answer to your question, See the "Amendment" to my question. |
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Apr 28 |
awarded | ● Nice Question |
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Apr 28 |
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Apr 27 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @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. |
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Apr 27 |
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Apr 27 |
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Hahn’s Embedding Theorem and the oldest open question in set theory 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. |
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Apr 27 |
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Hahn’s Embedding Theorem and the oldest open question in set theory @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. |
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Apr 27 |
awarded | ● Student |
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Apr 27 |
asked | Hahn’s Embedding Theorem and the oldest open question in set theory |
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Apr 18 |
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Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved? Since my comment is too long to for a comment, I have provided an extended comment in a separate answer. |
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Apr 18 |
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Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved? Vladimir: My response to the portion of your remark regarding Keisler is in the edited version of my answer. I will not respond to the nature of your remark regarding my work. |
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Apr 18 |
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Apr 17 |
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Apr 17 |
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Apr 17 |
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Apr 17 |
answered | Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved? |
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Apr 13 |
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Apr 11 |
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Surreals and NSA: some foundational issues Vladimir: I'll look into this for you and get back to you, though I fear it may be a day or two. |
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Apr 11 |
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Surreals and NSA: some foundational issues My pleasure, Vladimir. Glad to be of help. |
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Apr 11 |
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Surreals and NSA: some foundational issues Vladimir: I'm delighted to see that you edited your remarks on Hausdorff, presumably in light of my answer. By the way, the intimate relation between Hausdorff's pantachie and the surreal numbers is given by Theorem 17 of my BSL paper named in the answer. |
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Apr 11 |
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Surreals and NSA: some foundational issues (Continued from above): By the way, most of the nonstandard analysts I have spoken with (including H.J. Keisler, David Ross and Mauro Di Nasso, ...) are great fans of the surreals and would welcome the discovery of useful connections. |
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Apr 11 |
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Apr 11 |
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Surreals and NSA: some foundational issues David, don't despair, all of nonstandard analysis can be done using the surreals or some particular initial subfield of the surreals as the underlying ordered field in a nonstandard model of analysis. On the other hand, given the present state of knowledge, it would be inconvenient to do so. After all, there are highly visible nonstandard models of analysis that are easy to work with. Of course, none of them have the intuitive appeal of the surreals. Things may change with time, but the importance of the surreals has little to do with its relation to the great theory of nonstandard analysis. |
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Apr 11 |
answered | Surreals and NSA: some foundational issues |
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Apr 9 |
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dense orders are saturated @Ali: Oops, thanks, I've corrected it. I've also taken this opportunity to add a statement of the relevant part of Lemma 2 from the given paper. |
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Apr 9 |
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Apr 9 |
awarded | ● Commentator |
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Apr 9 |
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Apr 9 |
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dense orders are saturated @Ali (and Vladimir): It had already been demoted to an exercise in the 1973 edition of Chang and Keisler's Model Theory. There one is asked to prove that a real-closed field is an $\eta_{\alpha}$-field if and only if it is $\omega_{\alpha}$-saturated. Moreover, it is almost certain they expected one to prove it using familiar algebraic theorems regarding real-closed fields. They also assert earlier in the chapter, after proving the analog for ordered sets, that a proof like the one used by Erdös et al can be used to prove the result for real-closed fields. |
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Apr 9 |
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Apr 8 |
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dense orders are saturated As a footnote to my lengthy answer below, I note that when Simpson wrote the above and related remarks in 1999, I wrote to him and provided him with one of my papers on surreal numbers. In response, he wrote back: "Thanks very much, I have received your paper. It does indeed clarify some issues that came up in the FOM discussion of Conway. Regards, -- Steve Simpson" In response to a subsequent letter from me on the same matter, he wrote: "Thank you! Could you please send me a copy of your forthcoming paper in JSL? Welcome to FOM! -- Steve Simpson." Sorry, but I just remembered the exchange. |
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Apr 8 |
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dense orders are saturated Yes, Ali. In fact, the papers of Morely and Vaught and Alling, as well as the closely related works of Jónnson (on homogeneous-universal structures) and Keisler and Kochen all appeared about the same time. All of these folks were well aware of Hausdorff's work on $\eta_{\alpha}$-orderings, but not his construction of an $\eta_{\1}$-field. All of this work, which I believe deserves to be better well known, is part of the subject matter of my aforementioned historical work in progress. |
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Apr 8 |
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