In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ with the bounded value property, by which I mean: for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there exists $B$ in $F$ such that $-B \leq f(x) \leq B$ for all $x$ in $[a,b]_F$. (Here we say $f$ is continuous if it satisfies the usual $\epsilon,\delta$ definition of continuity, where all quantification is over $F$.)

In the absence of $AC_\omega$, what can one prove? E.g., could we perhaps prove the assertion using an explicit subfield of the Field of surreal numbers, such as the set of surreal numbers created prior to day $\omega_1$? (I'm not a logician, so it's possible that such notions as "the set of surreal numbers created prior to day $\omega_1$" intrinsically depend on $AC_\omega$ in ways I'm not seeing.)

In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ with the bounded value property, by which I mean: for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there exists $B$ in $F$ such that $-B \leq f(x) \leq B$ for all $x$ in $[a,b]_F$. (Here we say $f$ is continuous if it satisfies the usual $\epsilon,\delta$ definition of continuity, where all quantification is over $F$.)

In the absence of $AC_\omega$, what can one prove? E.g., could we perhaps prove the assertion using an explicit subfield of the Field of surreal numbers, such as the set of surreal numbers created prior to day $\omega_1$? (I'm not a logician, so it's possible that such notions as "the set of surreal numbers created prior to day $\omega_1$" intrinsically depend on $AC_\omega$ in ways I'm not seeing.)