Can a real-closed field of uncountable cofinality have a countable gap?

Question

Let $\mathbb{F}$ be a real-closed field with $\mathrm{cof}(\mathbb{F})\geq\omega_1$.

For subsets $A$ and $B$ of $\mathbb{F}$, we say that $\langle A, B \rangle$ is a pregap if $\forall a \in A \ \forall b \in B \ a < b$ and is a gap if it is a pregap and $\lnot \exists c \in \mathbb{F} \ \forall a \in A \ \forall b \in B \ a<c<b $ (we say $c$ is an interpolation of $\langle A, B \rangle $ ).

A gap $\langle A, B \rangle$ is a $(\kappa, \lambda)$-gap if $\mathrm{cof}(A) = \kappa$ and $\mathrm{cof}(B) = \lambda$, and is countable gap if $\mathrm{cof}(A), \mathrm{cof}(B) \leq \omega$.

Is it possible for $\mathbb{F}$ to have a countable gap?

It seems to be able to interpolate any countable pregap, but I have no idea to construct an interpolation.

Answer 1

Yes, such a field $\mathbb{F}$ can have a countable gap. Let $\kappa$ be any ordinal of uncountable cofinality, let $K$ be the field $\mathbb{Q}(S)$ where $S=\{x_\alpha\}_{\alpha<\kappa}$ is a set of $\kappa$ indeterminates, and order $K$ by saying that each $x_\alpha$ is greater than every element of the subfield generated by $\{x_\beta\}_{\beta<\alpha}$. Let $\mathbb{F}$ be the real closure of $K$, a real-closed field of cofinality $\operatorname{cof}(\kappa)\geq\omega_1$. I claim that $\mathbb{F}$ has a countable gap.

Specifically, let $A=\mathbb{N}$ and let $B=\{x_0^{1/n}:n\in\mathbb{Z}_+\}$. Clearly $(A,B)$ is an $(\omega,\omega)$-pregap. Suppose $c\in\mathbb{F}$ interpolates $(A,B)$. Observe that since every power of $c$ is less than $x_0$, each $x_\alpha$ is greater than every element of the subfield generated by $\{c\}\cup\{x_\beta\}_{\beta<\alpha}$. Since $c$ is greater than every integer, this implies that $\{x_\alpha\}_{\alpha<\kappa}\cup\{c\}$ is algebraically independent (since for any two distinct monomials, one is infinitely larger than the other). But by definition, every element of $\mathbb{F}$ is algebraic over the subfield generated by $\{x_\alpha\}_{\alpha<\kappa}$, so this is a contradiction. Thus no such $c$ exists and $(A,B)$ is an $(\omega,\omega)$-gap in $\mathbb{F}$.