= Solution
<Well-orders> are not first-order <axiomatisable>. Suppose a theory $T$ axiomatized them. Enlarge the language by constants $c_0,c_1,\ldots$ and add
$$
c_{i+1}<c_i\qquad(i\geq0).
$$
Every finite subset of the resulting theory has a model in the well-order $(\mathbb N,<)$: assign its finitely many constrained constants to a sufficiently long finite descending chain. Thus the <compactness theorem> supplies a model of $T$ with an infinite descending chain. The set of the $c_i$ has no least element, contradicting well-ordering. This shows why the second-order assertion that every nonempty subset has a least member cannot be replaced by a first-order theory in the poset language.
$$
\boxed{\text{Well-orders are not first-order axiomatizable.}}
$$
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