In model theory, two structures $\mathfrak{A}, \mathfrak{B}$ of identical signature $\Sigma$ are said to be *elementarily equivalent* ($\mathfrak{A} \equiv \mathfrak{B}$) if they satisfy exactly the same first-order sentences w.r.t. $\Sigma$. An astounding theorem giving an algebraic characterisation of this notion is the so-called Keisler-Shelah isomorphism theorem, proved originally by Keisler (assuming GCH) and then by Shelah (avoiding GCH), which we state in its modern strengthening (saying that only a single ultrafilter is needed):


$\mathfrak{A} \equiv \mathfrak{B} \ \iff \ \exists \mathcal{U} \text{ s.t. } (\Pi\_{i\in\mathcal{I}} \ \mathfrak{A})/\mathcal{U} \cong (\Pi\_{i\in\mathcal{I}} \ \mathfrak{B})/\mathcal{U},$


where $\mathcal{U}$ is a non-principal ultrafilter on, say, $\mathcal{I} = \mathbb{N}$. That is, two structures are elementarily equivalent iff they have isomorphic ultrapowers.


My question is the following (admittedly rather vague): Does anyone know of constructions in which an ultrafilter is **chosen** by an appeal to this characterisation and then used for other means? An example of what I have in mind would be something like this (using the fact that any two real closed fields are elementarily equivalent w.r.t. the language of ordered rings): In order to perform some construction $C$ I ``choose'' a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ by specifying it as a witness to the following isomorphism induced by Keisler-Shelah:


$\mathbb{R}^\mathbb{N}/\mathcal{U} \cong \mathbb{R}\_{alg}^\mathbb{N}/\mathcal{U},$ 


where $\mathbb{R}\_{alg}$ is the field of real algebraic numbers. So the construction $C$ should be dependent upon the fact that $\mathcal{U}$ is a non-principal ultrafilter bearing witness to the Keisler-Shelah isomorphism between some ultrapower of the reals and the algebraic reals, resp.


Also, a follow-up question: Let's say I'd like to ``solve'' the above isomorphism for $\mathcal{U}$. Are there interesting things in general known about the solution space, e.g., the set of all non-principal ultrafilters bearing witness to the Keisler-Shelah isomorphism for two fixed elementarily equivalent structures such as $\mathbb{R}$ and $\mathbb{R}\_{alg}$? What machinery is useful in investigating this?



Ultrafilters arising from Keisler-Shelah ultrapower characterisation of elementary equivalence
Under the Continuum Hypothesis, your solution space is *all* nonprincipal ultrafilters. This is because under CH, the ultrapower $M^N/U$ of a mathematical structure $M$ of size at most continuum does not actually depend on the (nonprincipal) ultrafilter $U$. One can see this by using the fact that the ultrapower will be saturated, and so one can run a back-and-forth argument to achieve the isomorphism. In particular, it follows under CH that any $U$ will witness your desired isomorphism for $R^N/U\cong (R\_{alg})^N/U$. 
(See Corollary 6.1.2 in Chang-Keisler's book Model Theory.) 


A similar fact holds for larger cardinals and larger structures under GCH, but here, one needs an additional assumption on the ultrafilter. Namely, Theorem 6.1.9 in Chang-Keisler asserts that if $2^\alpha=\alpha^+$ and $A$ and $B$ are two structures of size at most $\alpha^+$, then they are elementarily equivalent if and only if $\Pi\_DA\cong\Pi\_D B$ for any $\alpha^+$-good incomplete ultrafilter $D$ on $\alpha$. The proof uses the same saturation idea, and this establishes the Keisler-Shelah theorem in the case that GCH holds. 


Chang-Keisler states (page 393-394) that it is open whether the assertion of Theorem 6.1.9 stated above holds under $\neg CH$. 


