Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 16G / iv / Solution

The class of well-orderings is not first-order axiomatizable. If $S$ axiomatized it, add constants $c_0,c_1,\dots$ and sentencesEvery finite subset has a finite well-order containing the required finite descending chain. By the compactness theorem, the whole set has a model; that model satisfies $S$ but contains an infinite descending chain, so it is not a well-order. This contradiction proves the claim, as summarized by well-orders are not first-order axiomatizable.