We work in ZFC.


Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$.


A field $E$ is called real-closed if it has unique ordering $x\geq 0$ iff $\exists\_{y\in E}\ x = y^2$ and all polynomials of odd degree have a root.


An ordered field $E$ is called an $\eta\_1$-field if for all countable $A, B\subseteq E$ such that $A < B$ there is $x\in E$ such that $A < v < B$.


All hyperreal fields are real-closed $\eta\_1$-fields.


We have the following result:


**Theorem.** Assume CH. Then all real-closed $\eta\_1$-fields of size continuum are isomorphic.


**Corollary.** Assume CH. Then all hyperreal fields of size continuum are isomorphic.


Let P be "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic".


I'm wondering if one can get some insight into what kind of statement is P in relation to known set-theoretic principles.


Is P independent of ZFC+$\lnot$CH? Above theorem shows that ZFC+CH implies P.


Note: The choice of $X = \mathbb{R}$ was arbitrary, to not consider *every* possible topological space $X$ such that $C(X)/M$ is a hyperreal field of size continuum. If this is helpful, please consider a different space $X$ instead.



Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
In [On ultra powers of Boolean algebras](http://topology.nipissingu.ca/tp/reprints/v09/tp09205.pdf) (Topology Proceedings 9 (1984) 269-291) Alan Dow proved (Corollary 2.3) that $\neg\mathsf{CH}$ implies there are two fields of the form $C(\mathbb{N})/M$ that are not isomorphic (as ordered sets, and hence as ordered fields).


To be a bit more precise: Alan constructed two ultrafilters $u$ and $v$ on $\mathbb{N}$ with non-isomorphic ultrapowers. These yield maximal ideals $M\_u$ and $M\_v$ in $C(\mathbb{N})$ whose quotients are (isomorphic to) the respective ultrapowers. And because $\mathbb{N}$ is a closed discrete subspace of the metric space $\mathbb{R}$ these determine maximal ideals $N\_u$ and $N\_v$ in $C(\mathbb{R})$ whose quotients are isomorphic to those by $M\_u$ and $M\_v$ respectively.


