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.



an application of compactness theorem
You just add to $\Sigma$ the assertions $c\_{n+1}<c\_n$, where these are new constant symbols. Any finite collection of the resulting theory is consistent, since there are well-orders with a finite descending sequence of any particular finite length, but there can be no well-order realizing all of these statements. So this would violate the compactness theorem if $\Sigma$ were true in all and only the well orders. 


