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I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.

  1. Is there a soft model-theoretic construction showing that there must be such a field? I can imagine an argument using the nice model-theoretic features of real-closed fields, such as quantifier-elimination etc. (And if not, what is the best argument? Formal power series were mentioned, so I guess something like Hahn series.)

  2. Is the minimal countably saturated real-closed field also least, in the sense that it embeds into all other such fields?

  3. Is it unique? (I would be amazed, but this would be very welcome as a canonical structure.)

  4. Is the minimal countably saturated real-closed field simply $\text{No}_{\omega_1}$, that is, the surreal numbers born at any countable ordinal birthday? This field seems to be constructed in a minimal-like manner so as to be countably saturated, and so I would easily believe it is the one, if there is one. (Or perhaps it is one of several?)

I would appreciate any elucidation of these and related matters.

Meanwhile, I am aware that under CH, all smallest-size countably saturated real-closed fields are isomorphic, and furthermore, that this is equivalent to the continuum hypothesis. This fact was a central theme of my recent paper, How the continuum hypothesis could have been a fundamental axiom. So this question is mainly about the pure ZFC result, which is difficult only in the not CH case. In general, without CH, there are many non-isomorphic countably saturated real-closed fields of size continuum.

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    $\begingroup$ For clarity let me say that by "countably saturated" I intend what would also be called $\aleph_1$-saturated. $\endgroup$
    – Joel David Hamkins
    Commented Jul 12 at 2:36

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In Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de $\mathcal{C}(K)$, Acta Math. Acad. Sci. Hungar. 30 (1977), no.1-2, 113–127 (EuDML), Jean Esterle showed there is a least $\aleph_1$-saturated real-closed field in your sense of least. He further shows It can be characterized in terms of an $\aleph_1$-restricted Hahn field. Moreover, it is isomorphic to ${\bf No}(\omega_1)$ as you suggest. For a proof of the latter result, see the proof of Theorem 17 of my The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small (Bull. of Sym. Log. 18 (2012), pp. 1-45, doi:10.2178/bsl/1327328438, Project Euclid). However, since it properly contains an isomorphic copy of itself, it's not minimal, nor do I believe there is such a structure.

I am not aware of a model-theoretic proof of these results.

Edit (7/12)

In the same paper, Esterle shows that all $\aleph_1$-saturated real-closed fields of power $2^{\aleph_0}$ are isomorphic if and only if CH. Hence, if CH, we have uniqueness for least structures. As far as I know, the question of uniqueness without CH is open. On the other hand, I have wondered (but have never proved) if we limit ourselves to initial subfields of the surreals (i.e. subfields the predecessors of whose members in the structure considered as a subtree of the surreals coincide with the predecessors of its members in the surreal number tree), then ${\bf No}(\omega_1)$, which is initial, is the unique least initial $\aleph_1$-saturated real-closed subfield of the surreals. By a result from my Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers (J. Symb. Log. 6 (2001), pp. 1231–1258), every real-closed field is isomorphic to an initial subfield of the surreals. I'm pretty sure that (in ZFC) ${\bf No}(\omega_1)$ does not properly contain an initial isomorphic copy of a $\aleph_1$-saturated real-closed field.

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    $\begingroup$ Thanks for the answer! (And in fact I was thinking of writing to you to call your attention to my question.) Evidently "minimal" was being used with the sense of "least", namely, that it embeds into all others, which is an even better situation. But you don't specifically address the uniqueness question, except by saying "it is isomorphic to $\text{No}(\omega_1)$," but are they all definitely isomorphic? That is, is this a categorical characterization? $\endgroup$
    – Joel David Hamkins
    Commented Jul 12 at 4:01
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    $\begingroup$ I'll address the issue of uniqueness after a night's sleep. $\endgroup$
    – Philip Ehrlich
    Commented Jul 12 at 4:11
  • $\begingroup$ I believe that uniqueness can be proved by stratifying the field as an $\omega_1$ union of fields that have only countable cofinality cuts, and then undertaking a back-and-forth argument. This is like realizing $\text{No}(\omega_1)$ as the union of $\text{No}(\alpha)$ over countable ordinals $\alpha$. These smaller fields have all cuts of cofinality $\omega$. $\endgroup$
    – Joel David Hamkins
    Commented Jul 12 at 12:44
  • $\begingroup$ I saw your comment after I posted my edit. I'll have to give your comment some thought. $\endgroup$
    – Philip Ehrlich
    Commented Jul 12 at 14:23
  • $\begingroup$ I heard the back-and-forth argument from Woodin, who said it figured in his work on the Kaplansky conjecture. $\endgroup$
    – Joel David Hamkins
    Commented Jul 12 at 14:33

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