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 .

Simpson comments as follows:

"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.

Vladimir Kanovei

**After reading the comments**

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

Vladimir Kanovei

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.