In my answer to the related question, $AC_{\omega}$ was **only** used to ensure that one can get hold of a regular uncountable cardinal (i.e., $\omega_1$). And of course Gitik's remarkable theorem assures us that, assuming the consistency of a proper class of strongly compact cardinals, there is a model of $ZF$ with no uncountable regular cardinals.

But despite the limitations imposed by Gitik's theorem, we can construct, in $ZF$ alone, a proper class field $F$ such that the statement "$F$ has the bounded value property" is provable in $NBG$ (i.e., von-Neumann-Bernays-Gödel theory of classes).

**Explanation:** $NBG$ is a "conservative" extension of $ZF$, designed to handle "large objects" such as the class **V** of sets, the class **Ord** of ordinals, and the field **No** of surreal numbers. $NBG$ can prove that the class of ordinals **Ord** is regular in the sense that every function from **Ord** to **Ord** with bounded range is constant on an unbounded subclass of **Ord**. On the other hand, Schmerl's proof of Sikorski's theorem, when implemented in $NBG$ shows that for any regular uncountable cardinal $\kappa \leq$ **Ord** there is an ordered field $F$ of cardinality and cofinality $\kappa$ that satisfies $BW(\kappa)$. Moreover, The analysis of Schmerl's proof reveals that $F$ can be arranged to be *well-orderable*, which, coupled with THE LEMMA established in my answer, shows that $NBG$ proves that $F$ has the bounded value property.

I will close this note by relating the discussion to surreals **No**.

In the presence of $CH$ (the continuum hypothesis), and $AC$, every ordered field of cardinality at most $\aleph_1$ is isomorphic to a subfield of **No**($<\omega_1$), where **No**($<\omega_1$) is the collection of surreals "born" before $\omega_1$. Coupled with my answer to the other question, this shows that in the presence of $AC+CH$, we have:

**(1)** There is a subfield of **No**($<\omega_1$) with the bounded value property.

Moreover, using the "resplendence" property of saturated models, one can show:

**(2)** $ZFC+CH$ proves that **No**($<\omega_1$) does NOT have the bounded value property.

**(3)** $NBG$ plus global choice proves that **No** does NOT have the bounded value property.

My proofs of (2) and (3) using resplendence are nonconstructive, but there might be explicit failures of the bounded value property for **No** and **No**($<\omega_1$) already in $ZF$.