# dense orders are saturated

In a FOM msg of Fri May 21 19:59:44 EDT 1999, http://www.cs.nyu.edu/pipermail/fom/1999-May/003149.html Simpson gave a short proof of the following theorem:

Theorem. Let F be a real closed ordered field and suppose that the ordering of F is $\kappa$-dense, i.e., for every pair of sets $X,Y$ in F with $X < Y$ and ${\rm card}(X \cup Y)\le \kappa$, there exists z in F such that $X < z < Y$. Then F is ($\kappa^+$)-saturated.

In other words, the saturation property wrt cuts implies the full model-theoretic saturation.

It is an immediate corollary that the known field surreal numbers NO is ordered field-isomorphic to a class-size full set-saturated elementary extension R* of the reals R (under the global choice axiom), and hence NO admits a full superstructure which is also an elementary extension of the full superstructure over R .

"This result [the thm above - VK] may be well known to those who know it well. But I didn't know it before. Can anyone supply a reference for it? "

As far as I can follow the FOM thread, nobody managed to give a reference. Neither can I now. Therefore, a question: will anybody be so kind as to offer a reference wrt the theorem above in a published paper.

First of all, I sincerely thank correspondents for taking time and trouble with this issue.

In the rest,

1) Regarding [Erdos ea 1955] - thankfully it is available from the Erdos site - theorem 2.1 there literally says that:

for any ordinal $\alpha>0$, any two rcf of cardinality $\aleph_\alpha$ which are $\eta_\alpha$ sets are isomorphic.

There is no saturation claim here, so one has still to work towards saturation from this result on. Yet most importantly the setup is restricted to the case of rcf of cardinality $\aleph_\alpha$ - which (and Erdos ea note it just after the theorem) is equivalent to CH being true on $\aleph_\alpha$. To conclude, [Erdos ea 1955] is not a solution to the problem.

2) Regarding [Marker, Model theory] - thankfully, it's downloadable either. Ex 4.5.18 on p. 165 claims that a rcof F is $\kappa$-saturated iff the underlying order is, thus leaving aside the (how much simpler?) problem to prove the Simpson's theorem for orders (or just some type of orders) instead of fields.

3) Regarding [Ehrlich] pre-1999 papers mentioned - thankfully, downloadable from Philip's www page. The AU 1988 paper contains a theorem on p. 12 saying that

if $\aleph_\alpha$ is a saturation cardinal then a certain $\alpha$-fragment of No is a unique $\aleph_\alpha$-universally extending ordered field of cardinality $\aleph_\alpha$.

Even if we take it as given that the universally extending property implies saturation in this context, this does not yield a proof of Simpson's theorem in its general context.

With thanks again, I reproduce here Simpson's own proof, just for the sake of completeness of the discussion.

Proof. By Tarski's result on quantifier elimination for real closed ordered fields, any subset of $F$ which is definable over $F$ allowing parameters from $F$ is a finite union of intervals, all of whose endpoints are in $F$. But then Tychonoff's theorem [in a later msg, Simpson refers to the Rado selection theorem, which is better - VK] plus $\kappa$-density of $F$ implies that any family of $\kappa$ such sets has the finite intersection property. Hence $F$ is $\kappa^+$-saturated.$\square$

Best regards

PS. Regarding Hausdorff, his early papers on ordered sets have been reprinted in a brand-new Band 1a of Felix Hausdorff, Gesammelte Werke, with comments (including those on "Pantachies"), now available.

A commented English translation of Hausdorff early papers, see Hausdorff on ordered sets, Volume 25 of History of Mathematics, Felix Hausdorff, Editor J. Jacob M. Plotkin, AMS, 2005.

• 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. – Philip Ehrlich Apr 8 '13 at 21:38
• @Vladimir: you are right that the statement of Theorem 2.1 of the paper by Erdős et al. does not yield the result you are interested in. However, the proof of Theorem 2.1 is carried out by a back-and-forth argument, a key step of which involves the key idea of reducing the realizability of a type in the language of ordered fields, to a purely order-theoretic type, which is precisely what is needed to prove the result you are interested in (note that this proof does not invoke Tarski's elimination of quantifiers in real closed fields)--(continued in the next comment). – Ali Enayat Apr 9 '13 at 18:20
• My guess is that after the appearance of the paper of Erdős at al., the result you are interested in became "common knowledge" among the cognoscenti; which is corroborated by the fact that, similar to many venerable theorems, it was "demoted" to an exercise in Marker's text. – Ali Enayat Apr 9 '13 at 18:25
• @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. – Philip Ehrlich Apr 9 '13 at 19:38
• Indeed. In my CK 1990 p. 343 Prop 5.4.2 claims that dense $\eta_\alpha$ orderings w/o endpoints are $\aleph_\alpha$-saturated. The full theorem (for rcof) is Ex 5.4.5 on p. 369, given w/o hints and w/o comments in the Historical notes chapter. Still, I think Simpson's proof stands for its elegance and brevity. Thanks for the comment. – Vladimir Kanovei Apr 10 '13 at 3:53

Philip Ehrich and Emil Jeřábek have given useful references (and Dave Marker and I seem to have simultaneously written our responses).

This note is to point out that the result itself was originally established by Erdös, Gillman, and Henriksen; it appears as Theorem 2.1 of their paper below:

Erdös, P.; Gillman, L.; Henriksen, M. An isomorphism theorem for real-closed fields. Ann. of Math. (2) 61, (1955). 542–554.

By the way, an analgous result is also true, surprisingly, for models of PA, and of ZFC.

For PA, it is due to Jean-François Pabion, in his paper:

Saturated models of Peano arithmetic. J. Symbolic Logic 47 (1982), no. 3, 625–637.

For ZFC, it is due to Schmerl and Kaufmann; see line 4, section 2 of the following paper (which also includes a proof of Pabion's result).

Kaufmann, M; J.H. Schmerl, Saturation and simple extensions of models of Peano arithmetic. Ann. Pure Appl. Logic 27 (1984), no. 2, 109–136.

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

Vladimir: 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)."

I then go on to prove as part of Lemma 2: Let 0 < $\alpha$ < or = On. S is an $\aleph_{\alpha}$- universally extending ordered field if and only if S is a real-closed field that is an $\eta_{\alpha}$-ordering.

[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, thanks for the historical tidbits in your emendation; I was not aware of the work of Hausdorff, nor of the connection between the work of Alling and that of Erdös et al. It is amusing that Alling's paper came out in the same year as the classical paper of Morley and Vaught on saturated models (Homogeneous universal models. Math. Scand. 11 1962, 37–57). – Ali Enayat Apr 8 '13 at 17:37
• 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. – Philip Ehrlich Apr 8 '13 at 19:02
• @Philip: in the second emendation; Vladmir's name has morphed to Victor. – Ali Enayat Apr 9 '13 at 20:02
• @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. – Philip Ehrlich Apr 9 '13 at 22:46

I think the main idea goes back to the following paper.

Erdös, P.; Gillman, L.; Henriksen, M. An isomorphism theorem for real-closed fields. Ann. of Math. (2) 61, (1955). 542–554.

This characterization for dense linear orders can be found as Proposition 16.1 in Sachs’ Saturated model theory. Note that he defines $\kappa$-density differently than you (this property is also known as orders of type $\eta_\alpha$ in the literature, by the way). I can’t find a discussion of saturated real-closed fields in the book, but the fact that a rcf is $\kappa$-saturated iff its underlying order is is included e.g. as Exercise 4.5.18 in Marker’s Model theory: an introduction.