Skip to main content
4 of 6
added 1940 characters in body
Philip Ehrlich
  • 6.5k
  • 1
  • 42
  • 37

Vladimir,

The results relating saturated real-closed fields and No go back to the following two papers of mine.

“Absolutely Saturated Models,” Fundamenta Mathematicae, 133 (1989), pp. 39-46.

“An Alternative Construction of Conway's Ordered Field No,” Algebra Universalis, 25 (1988), pp. 7-16. Errata, Ibid. 25, p. 233.

Moreover, in the 1988 paper (which was submitted in 1982, but did not appear till 1988!) I also mention in passing that certain kappa-saturated initial subfields of No are isomorphic to the underlying ordered fields in nonstandard models of analysis. Furthermore, in the following two works of mine (working in NBG with global choice), the fact that the full field of surreal numbers is isomorphic to the underlying ordered field in the (up to isomorphism unique) On-saturated hyperreal number system is stated and established, respectively.

“The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small,” in Philosophical Insights into Logic and Mathematics (Abstracts), Université de Nancy 2, Laboratoire de Philosophie et d’Historie des Sciences, Archive Henri Poincaré, Nancy, France, 2002, pp. 41-43.

“The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45.

Although I was aware of the result for some time before 2002, it was only after an exchange of letters between H.J. Keisler and myself on the matter in 2002 that led me to put it into print.

By the way, the relation between kappa-saturated real-closed fields and real-closed fields that are kappa-dense (in your sense) goes back to the 1960's and is due to H.J. Keisler and (to a lesser extent) Simon Kochen.

I hope this helps.

Emendation

Dave’s and Ali’s references to the important paper by Erdös, Gillman, and Henriksen motivates the following emendation to my original response. As I point out in Section 8 of my aforementioned BSL paper, the existence of a real-closed field that is an $\eta_{1}$-ordering was first established by Hausdorff in his “Die Graduierung nach dem Endverlauf”, Abhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch-Physische Klasse, vol. 31, pp. 295–335 (1909). Writing before Artin and Schrier [1926], Hausdorff of course does not refer to his field as real closed, but he essentially establishes it is real-closed by showing it is the union of a chain of ordered fields, each of which admits no algebraic extension to a more inclusive ordered field. In addition to not being aware of Hausdorff’s paper, Erdös, Gillman, and Henriksen were not aware if there are real-closed fields that are $\eta_{\alpha}$-orderings for $\alpha$ greater than 1. In response to their question to that effect, in 1962 the order-algebrist Norman Alling showed there is (up to isomorphism) a unique real-closed field that is an $\eta_{\alpha}$-ordering of power $\aleph_{\alpha}$ in each saturation cardinal (On the existence of real-closed fields that are $\eta_{\alpha}$-sets of power $\aleph_{\alpha}$, Transactions of the American Mathematical Society, vol. 103, pp. 341–352 (1962)). The connection between real-closed fields that are $\eta_{\alpha}$-orderings and $\alpha$-saturated real-closed fields date from the same period.

All of the above is discussed in detail in a paper I am presently working on entitled “From du-Bois Reymond’s Infinitary Pantachie to the Surreal Numbers.”

Second Emendation

Victor: Just to be clear, the references to my pre-1999 papers were clearly not intended to supply the proof you sought. On the other hand, contrary to what you say, the general case is addressed in both of those papers. For example, on page 8 of my 1986 paper (with reference to classical works of Jónnson, Morley and Vaught, Chang and Keisler) I write:

"Following Jónnson ([16], p.149), a model A for a theory T in a language L is said to be $\kappa$-universally extending if for any models B and C of T in L where B is a substructure of A, C is an extension of B, and |B|, |C| < $\kappa$, there is a model C’ of T in L that is a substructure of A and an isomorphism from C onto C’ extending the identity map on B. When, as in the case of ordered fields, T is a Jónnson theory, i.e. an inductive first-order theory having the amalgamation and joint embedding properties as well as an infinite model (cf. [18], p.147), the $\kappa$ -universally extending models of T coincide with the models of T that are $\kappa$-homogeneous and $\kappa$-universal with respect to models of T (cf. [24], Corollary 2.5e and the remarks on p. 153 of [18]). For the case at hand, of course, the latter structures are the $\kappa$-saturated models for the theory of real-closed ordered fields, i.e., the real-closed fields that are $\eta_{\alpha}$-orderings for $\alpha$ > 0 (cf. [7], Ch. 5)."

[7] Chang, C. C. and Keisler, J. H., Model Theory, North-Holland, 1973.

[16] Jónnson, B. “Homogeneous universal relational systems”, Math. Scand. 8 1960 137–142. [1

[18] Jónnson, B. “Extensions of relational structures” 1965 Theory of Models (Proc. 1963 Internat. Sympos. Berkeley) pp. 146–157 North-Holland, Amsterdam.

[24] Morley, M. and Vaught, R. “Homogeneous universal models”, Math. Scand.11 1962 pp. 37–57.

Best of luck with your search.

Philip Ehrlich
  • 6.5k
  • 1
  • 42
  • 37