It's known that the class of well-founded linear orders is not axiomatizable in FO logic. But I can't understand the relation between the aforesaid argument and the compactness theorem.
Let $\mathcal{A}$ be a well-founded linear order (which means it contains no infinite descending chain for $a_1, a_2, ...$ such that $... <^{\mathcal{A}} a_3<^{\mathcal{A}} a_2<^{\mathcal{A}} a_1$). To show the truthfulness of the stated argument, one can say that there is no set of sentences like $\Sigma$ such that:
$\mathcal{A} \models \Sigma \Leftrightarrow \mathcal{A}$ is well-founded.
How can one use compactness theorem for this case? Any reference will be welcome.