I am working on a metalogic problem, as follows:
Show that there is no theory $\Sigma$ in the language of ordering such that for all models $(A, \leq)$ in this language, one has that $(A, \leq) \mid= \Sigma$ $\textit{iff}$ $(A, \leq)$ is well-ordered.
I have worked out the proof to the point where what I need to show is that well-orderedness is not expressible by a set of first-order sentences. Searching around the internet has confirmed that this claim is in fact true, but I cannot figure out how to prove it.
This is a standard basic exercise in compactness, so here are some very strong hints. As with many proofs by compactness, try to expand your language first by adding infinitely many new constant symbols. Then proceed by contradiction; suppose you can express well-orderedness with a first-order theory T. Use these symbols to build the “right” set of sentences which you can show is consistent with your theory by compactness, which will contradict the fact that T’s models were all well-ordered.