What is the class of models of the $\forall$-theory of the class of well-ordered sets?

Question

I know that the class $W$ of well-ordered sets is not an axiomatizable class. However, it does have an associated first-order theory $Th(W)$. Now, consider the $\forall$-part of the theory $Th(W)$, which I denote by $\forall Th(W)$. What is the class of models of that theory? Certainly, that class of models is a subclass of the class $L$ of linearly ordered sets, but is it in fact equal to the class $L$? If not, can someone give an explicit description of the models of $\forall Th(W)$?

Answer 1

It is the class of linearly ordered sets. In fact, if $\sigma$ is a $\forall\exists$ first-order sentence (in particular if $\sigma$ is a $\forall$ sentence), the following statements are equivalent:
(1) $\sigma$ is true in every linearly ordered set.
(2) $\sigma$ is true in every well-ordered set.
(3) $\sigma$ is true in every finite linearly ordered set.

For the implication $(3)\implies(1)$ note that every set is a directed union of finite sets, and $\forall\exists$ sentences are preserved by directed unions. (Alternatively, $\forall\exists$ sentences are preserved by unions of chains, and a countable set is the union of a chain of finite sets, so $\sigma$ holds in every countable linearly ordered set, and therefore by Löwenheim–Skolem in every linearly ordered set.)